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### Idea of a Function

• A function is a relation between two sets: a first set and a second set. For each element from the first set, the function assigns precisely one element in the second set.
• Just like variables, it is useful to name functions with a symbol. Most often we will use f, but sometimes we'll use g, h, or whatever else makes sense.
• If we want to talk about what f assigns to some input x, we show this with f(x). The first symbol is the "name" of the function, and the second symbol in parentheses is what the function is acting on.
• There are many metaphors we can use to help us make sense of how a function works:
• Transformation: The function transforms elements from one set into another. From problem to problem, the "rules" for transformation will (usually) change as we use different functions. However, as long as we're using the same function, the rules never change: if we put in the same x, we will always get the same f(x) as our result.
• Map: The function is a "map"-it tells us how to get from one set to another set. Of course, if we start at a different place, we might end up at a different destination. However, the map itself never changes: if we start at the same place, we always arrive at the same destination.
• Machine: The function is a machine that "eats" inputs and then produces outputs. What it produces depends on what it eats, but the machine is reliable: if it eats the same thing, it always produces the same output.
• A very important idea is that, given the same input, a function will always produce the same output. If f(5) = 17 the first time, but then f(5) = 27 the second time, it is NOT a function. [However, there would be no problem with g(17) = 5 and g(27) = 5.]
• In math, most of the functions we'll work with (at least for the next few years) will take in real numbers and produce real numbers (ℝ→ ℝ).
• When we are given a function, we will usually be told what its "rule" is: how it maps inputs to outputs. For example, if f(x) = 7x − 5, its rule is, "Multiply 7 and x together, then subtract by 5." [It's important to notice how x acts as a placeholder: it tells us what happens to whatever the function acts upon.]
• If we want to evaluate a function at a specific value, we just apply the "rule" to whatever our input value is. In practice, this turns out to be pretty simple: usually we are given a formula for each function, so we just follow the method of substitution. Remember to wrap your substitution in parentheses, though! If you don't, you could make a mistake on a complex input.
 f(x) = 7x − 5    ⇒     f(a−3) = 7(a−3) − 5
• A good way to see the behavior of a function is by creating a table of values. On one side we have input values, while the other side shows us what the function outputs when given that input. Sometimes we'll be given the table, while other times we will have to decide what inputs to use and determine the outputs ourselves.
• The domain is the set of all inputs that the function can accept. While we generally assume that all of ℝ can be used as inputs, sometimes certain values will "break" our function: the output cannot be defined. Thus, our domain is all of ℝ except that which breaks our function. [For now, we mostly have to watch out for dividing by zero and taking square roots of negative numbers. Later in the course we'll also have to be careful about inverse trigonometric functions and logarithms.]
• The range is the set of all possible outputs a function can assign (given some domain). While these values will always be in ℝ (unless otherwise noted), they do not necessarily cover all of ℝ.

### Idea of a Function

Consider the function
 f(x) = 2x + 1.
Explain what the notation means and how to interpret it.
• In f(x), the first symbol (f, in this case) is the "name" of the function. It is a symbol that we are using to represent the function.
• The symbol in the parentheses (x, in this case) is what the function is acting on. It is what the function is taking as an input.
• What is on the right side of the equation is what the function f outputs when given the input x.
[Answers may vary] f is the "name" of the function, x is what the function is acting on (its input), and what is to the right of the equals sign is what the function outputs [we can also interpret the right side as a rule for how the function works].
Determine if the below relationship could be from a function:
 3
 →
 7
 1
 →
 −4
 −8
 →
 20
 3
 →
 10
 5
 →
 −4
• A function takes some input and produces some output. For this problem, the left side is the input, and the right side is the output the "function" transforms it into.
• To be a function, this transformation must be reliable. Given the same input, it must always produce the same output.
• It's alright for two different inputs to produce the same output (1 → −4 and 5 → −4 is fine, for example), the issue is when the same input produces two different outputs.
No, this relationship cannot be a function because 3 → 7 and 3 → 10: the same input goes to two different outputs, which a function is not allowed to do.
Let f(x) = 4x − 2. What is the value of f(3)?
• To evaluate f(3), plug it into the function.
• We do this by swapping the location of x for 3. [Because we get from f(x) to f(3) by substituting one for the other.]
 f(3) = 4(3) − 2.
10
Let g(k) = −2k2 + 3k −3. What is the value of g(−4)?
• To evaluate g(−4), plug it into the function.
• We do this by swapping the location of k for −4. [Because we get from g(k) to g(−4) by substituting one for the other.]
 g(−4) = −2 (−4)2 + 3 (−4) −3.
• Remember to work by the order of operations while simplifying: parentheses, exponents/radicals, multiplication/division, addition/subtraction.
−47
If h(x) = √{x−3}+2, fill in the table below:
 x
 h(x)
 3
 4
 7
 10
 12
• For each of the x-values, just plug it into the function like normal, then simplify.
• One of the inputs will not come out as an integer: you'll be left with a square root. All the others will come out as whole numbers, though.

 x
 h(x)
 3
 2
 4
 3
 7
 4
 10
 √7 + 2
 12
 5
If f(x) = 3x2 + 2x − 7, what is f(2x2)?
• To find out what f(2x2) is, we evaluate basically like normal. Swap out all of the original x-locations for x2.

•  f(x) = 3x2 + 2x − 7        ⇒        f(2x2) = 3(2x2)2 + 2 (2x2) − 7
• Simplify. [Remember that (2x2)2 = 2x2 ·2x2 = 4x4.]
f(2x2) = 12 x4 + 4x2 − 7
If M(k) = 2k2 + 3k, fill in the table below:
 k
 M(k)
 a
 5b
 −k
 x+3
• For each of the k-values, just plug it in to the function and simplify.
• When substituting in values, make sure to wrap them in parentheses ( ) so that everything is applied correctly.
• When plugging in x+3, you'll have to expand (x+3)2. You've probably learned this before (possibly called the FOIL method), but if you haven't we'll study it later in the section of this course on polynomials.
 (x+3)2 = (x+3)(x+3) = x2 + 3x + 3x + 9 = x2 + 6x + 9
• Remember, the 2 distributes to the entire expansion of (x+3)2, but only after it has been expanded.
 M(x+3) = 2 (x+3)2 + 3 (x+3)

 k
 M(k)
 a
 2a2 + 3a
 5b
 50b2 + 15b
 −k
 2k2 − 3k
 x+3
 2x2 + 15x + 27
What is the domain of this function:
 f(x) = 1 x + x2x+2 ?
• The domain of a function is all the values that can be used as inputs.
• The only possible issue that could come up in f(x) is if the denominator (bottom) of either fraction was 0, because dividing by 0 is not defined.
• Since dividing by 0 is not defined, if the denominator is 0, whatever x caused that can not be allowed.
• That means any x that would work in either of these equations
 x = 0               x+2 = 0
is not allowed.
• The two values of x that are not allowed are 0 and −2.
• Any other value for x than those two "forbidden values" is fine to put into the function, so everything else is in the domain.
The domain of f(x) is all x where x ≠ 0 and x ≠ −2.
Let g(x) = x2 −15. What is the range of g(x)?
• The range of a function is all the values that the function can produce as outputs.
• Notice that the lowest value x2 can possibly be is 0. [This is because any negative will turn positive when squared.]
• Since −15 never changes and the lowest value x2 can be is 0, the lowest possible value that g(x) can output is −15.
• Notice that x2 can create any value equal to 0 or larger.
• Since −15 never changes, by just adding an appropriately large x2, we can get to any value above −15.
The range of g(x) is [−15,  ∞).
Professor Erdös can produce one math paper for every five days he works on mathematics. Express the number of papers P that he writes in a year as a function of the number of days t that he works in that year.
What is the domain of P(t)? [Hint: think about the limitations on t.]
• P is the number of papers produced, t is the number of days worked.
• From the problem, we can create an equation that connects P and t. If Erdös produces one paper every five days, then we have
 P = t 5 .
• Since P is entirely dependent on t, we can express P as a function of t:
 P(t) = t 5 .
You plug a value of t in to P(t), and it gives how many papers he produced that year.
• Thedomain is all the possible values we can plug in for t and still have the function make sense. However, for it to make sense, it also has to make sense for t to be that value. Remember, in the problem it said that this was how many days that he works in the year. What does that limit t to?
P(t) = [t/5]. The domain is [0,  365] because t can not exceed the number of days there are in a year. (Although you could make an argument about leap years, if you really wanted to.)

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Idea of a Function

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Introduction 0:04
• What is a Function? 1:06
• A Visual Example and Non-Example 1:30
• Function Notation 3:47
• f(x)
• Express What Sets the Function Acts On
• Metaphors for a Function 6:17
• Transformation
• Map
• Machine
• Same Input Always Gives Same Output 10:01
• If We Put the Same Input Into a Function, It Will Always Produce the Same Output
• Example of Something That is Not a Function
• A Non-Numerical Example 12:10
• The Functions We Will Use 15:05
• Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
• Usually Told the Rule of a Given Function
• How To Use a Function 16:18
• Apply the Rule to Whatever Our Input Value Is
• Make Sure to Wrap Your Substitutions in Parentheses
• Functions and Tables 17:36
• Table of Values, Sometimes Called a T-Table
• Example
• Domain: What Goes In 18:55
• The Domain is the Set of all Inputs That the Function Can Accept
• Example
• Range: What Comes Out 21:27
• The Range is the Set of All Possible Outputs a Function Can Assign
• Example
• Another Example Would Be Our Initial Function From Earlier in This Lesson
• Example 1 23:45
• Example 2 25:22
• Example 3 27:27
• Example 4 29:23
• Example 5 33:33

### Transcription: Idea of a Function

Hi--welcome back to Educator.com.0000

Today, we are going to talk about the idea of a function.0002

Functions are extremely important to mathematics: you have certainly encountered them before.0006

But you might not have fully understood how they work and what they are doing.0011

This lesson is here to give us a clear understanding of what it means for something to be a function, and how functions work.0015

Since functions are so important, they are going to come up in every single lesson you learn about in this course.0021

And they are going to come up in every single concept you talk about in calculus.0026

And they are going to keep coming up, as long as you are studying math.0028

Make sure you watch this entire lesson; it is so important to have a good, grounded, fundamental concept of what a function is,0031

because it is going to keep getting used in everything that we talk about.0038

This is probably the single most important lesson of this entire course,0042

because so many later ideas are going to talk about functions.0045

Also, it would help to have watched the previous lesson on sets, elements, and numbers,0049

because we are going to be talking about how sets are connected to functions.0054

So if you haven't done that, I would recommend that you go and watch that one first,0057

because it will help explain a lot of what we are talking about here, because functions are relying on the idea of sets.0060

All right, let's jump into it: what is a function?0065

A function is a relation between two sets: a first set and a second set.0069

For each element from the first set, the function assigns precisely one element in the second set.0074

So, we will point at some element in the first set, and it will say, "Here is an element from the second set."0080

Point at another element from the first set, and it will tell us, "Here is some element from the second set."0085

That is the idea of a function; here is a visual example for it.0089

We could have something where all of the squares are the first kind--it is our first set--and all of the round things on this side are our second set.0092

So, second would be the second column, and first set would be the first column.0101

We could have news get put onto paper; we say that news, the function, gives us paper.0106

We say that cheese, the function, gives us burger; we say that good, the function, gives us bye.0112

We say that sand, the function, gives us paper; we say that bubble, the function, gives us gum.0118

So, there are only five elements in our first set, and only four elements in our second set.0123

But this is a perfectly reasonable function: newspaper, cheeseburger, goodbye, sandpaper, bubblegum.0127

The only one you might be wondering about is..."Wait, news goes to paper and sand goes to paper."0134

There is no problem with that: we only said that the function has to give us something when we point at something in the first one.0138

We never said that it has to be a different thing, every single thing that we point to; it just has to give us something for it.0144

That is what we have here: we have something where everything that we call out on the first side...0150

we call out news, and in turn, it responds by telling us paper.0154

We call out good, and in return, it says bye; that is how it is working here with this function.0159

Here is a non-example: in this one, we say tree, but the function gives us four different possibilities.0167

Sometimes it gives out maple; but other times it gives out oak; but other times it gives out apple; but other times it gives out pine.0174

And then fruit--if we go to fruit, it sometimes gives out apple, and sometimes it gives out grape.0182

This isn't allowed, because it is only allowed to give one response to a given input.0188

We tell it one element from our first set; it can only tell us one element from the second set.0195

It is not allowed to give us a whole bunch of different choices to pick and choose from.0200

Sometimes it is going to be maple; sometimes it is going to be oak; sometimes it is going to be pine.0204

No; it has to be one thing, and one thing only.0207

That is what it requires to be a function; so this is not an example--this is not allowed, because we can't have it be multiple things coming out of this.0210

It has to only be that one input will only give us one output.0219

And as long as we keep putting in that same input, it can only give us the same output.0223

Just like variables, it is useful to name functions with a symbol; so let's talk about how notation works here.0228

Most often, the symbol we will use to talk about a function is f; but sometimes0233

we are also going to use g, h, or whatever else will make sense, depending on the context.0237

But often, we are going to end up seeing f.0242

If we want to talk about what f assigns to some input x, if x is the element in our first set,0245

if that is what we call the element in our first set that we use f on, then it will be assigned to "f of x,"0250

f acting on x--what f gives out when given x; so the first symbol is the name of the function that we are using;0257

then, the second symbol, in parentheses, is what the function is acting on.0267

So, f--the name of our function--is acting on x; and then, that whole thing together is f(x); f(x) is the name of what comes out of it.0276

So, f is the name of what is doing the acting; x (or whatever is in the parentheses)...the first symbol was the name0286

of whatever is doing the acting; the thing inside of the parentheses is the name of what is being acted on;0292

and then, the whole thing taken together is where we are when we use the function on that element--0298

what we get output to where we come to.0304

Now, there could be a little bit of confusion about f(x), because it is f, parenthesis, x.0307

And we know that parentheses...if I wrote 2(3), that would mean 2 times 3, right?0312

So, we might think f times x; but we are going to know from context that f is a function, and not something that we multiply.0319

So, when f is a function, we don't have to worry about using multiplication, if it is f on some element.0325

It is always going to be f of that element, never f times, unless we are talking about that explicitly.0330

But if it is just in parentheses, it is not going to be multiplication.0336

So, when you see parentheses, and it is a function, it isn't implying multiplication, like when we are dealing with numbers.0339

If we want to express what sets the function acts on, we can write f:a→b.0346

What this is: it is "f goes from a to b"; it takes elements from a, our first set; and then it assigns them elements from b.0353

Normally, it won't be necessary for us in this course (and probably for the next couple of years)--0363

it won't be necessary to name the sets that our function is working on.0366

But why that is, we will discuss later: it is going to be pretty simple, but we will discuss it later when we get to it.0371

There are a lot of metaphors that we can use to help us understand what is going on in a function.0377

Here are three metaphors to help us understand what happens when f takes things from a and goes to b.0382

Our first idea is transformation: the function transforms elements from one set into another.0388

It takes an element x, contained in a, and then it transforms it into an element in b, which we call f(x), or f acting on x.0394

f(x) is what it has been transformed into; that is what it is after the transformation.0406

Now, from problem to problem, the rules for transformation will usually change as we use different functions.0411

One function is generally going to have a different set of rules for how its function works than another function.0416

But if we are using the same function--if we are in the same problem, using the same function--0421

the rules never change: if we put in the same x, we will always get the same f(x) as our result.0425

The rules for how the transformation works are always the same.0431

So, if the same thing goes in, the same thing always comes out.0434

Another way we can look at it is a map: it tells us how to get from one set to another set.0438

It is sort of a guide, directions for how to get from one place to another place.0443

Of course, if we start at a different starting location, a different starting place,0449

different elements in a, we might end up at a different destination--different elements in b.0453

If I say, "Go 100 kilometers north," you are going to end up in totally different places0458

if you start in Mexico, if you start in California, if you start in England, if you start in South Africa, or if you start in Japan.0462

Each one of these places...if you start in Egypt...is going to end up going to a totally different place, even though they are all still the same direction.0469

You are still doing the same thing; you are still going 100 kilometers north in all of these cases.0476

But because you started in a different place, you end up in a different place.0481

So, a different starting place, a different element that we are acting on, a different element that we are mapping,0484

will normally cause us to have a different destination--a different place that we land on.0489

The math itself, though, never changes: if we start at the same place, we always arrive at the same destination.0495

So, if we start in San Jose, California, and then we go 100 kilometers to the north...I actually have no idea where that is.0501

But we will be 100 kilometers north of San Jose.0508

And then, if we start in San Jose again on another day, and we go 100 kilometers north, we are going to end up being in the exact same place.0510

And if we go to San Jose, and then we go 100 kilometers north again, we are going to end up being in the exact same place.0516

And people are probably going to wonder, "Why does this person keep showing up here?"0521

And that is because we are following the same map.0525

The directions, the transformation that the map gives us, the way we go, isn't going to change each time.0527

It only changes when we start from a new place.0533

Finally, one last way to visualize it is the idea of a machine.0537

We can visualize a function as a machine that eats elements from a, and it produces elements from b.0541

What it produces depends on what it eats, but the machine is reliable: if it eats the same thing, it always produces the same output.0547

For example, if we have x right here, and we push it into our machine, f, it goes into the machine;0553

then the machine works on it and crunches it, crunches it, crunches it; and it gives out f(x).0559

So, we are going from the set a to the set b.0564

Now, one thing about the machine is that it is perfectly reliable; the machine is reliable.0572

If it eats the same thing, it produces the same output.0577

If we put in x, it will always give out f(x); so the first time we put in x, it gives out f(x); the second time, f(x); the third time, f(x); the fiftieth time, f(x).0581

Just like when we started in San Jose, and we went 100 kilometers north, each time we always ended up coming to the same place;0590

you put the same thing into the machine; the same thing comes out.0595

This idea is so important; we are going to talk about it really explicitly.0599

We have said this one way or another for all of our different ways of thinking about functions.0602

But it is so important--it is such an important characteristic of functions--we want to make sure that we know it.0606

If we put the same input into a function, it will always produce the same output.0611

Now, the input and the output could be totally different; the input is not necessarily going to be where we show up in the output.0616

You start in San Jose, and then you show up in some farmer's field 100 kilometers to the north.0622

But you are going to come out to that same farmer's field each time, because you are showing up at the same location.0626

So, for a function to make sense and be well-defined, for it to work, its rules must never change.0633

For example, if f(2), f acting on 2, gives out 7; if f(2) equals 7 the first time, then f(2) = 7 the second time; and f(2) = 7 every time.0639

No matter how many times f operates on 2, no matter what, it is always going to give out the same thing.0650

That is what it means to be a function: your rules don't change when you are going on the same thing.0654

You work on one element the same way each time; you always map it; you always transform it; you always assign it to the same place.0660

Here is something that is not a function: g(cat) = fur, g(cat) = whiskers, g(cat) = quiet.0671

This can't be a function, because we have three totally different destinations when we plug in cat.0678

And what determines whether we go to fur, whiskers, or quiet?0683

There is no reason why we should use one set of rules or another set of rules, so it is not a function.0686

There is no reliability here; we don't know, when we plug in cat, if we are going to go to fur, whiskers, or quiet.0690

So, it is not a function; but we could have a function that was h(fur) goes to cat, h(whiskers) goes to cat, h(quiet) goes to cat.0695

It is not that there is a problem with having us land on the same place.0703

No matter what we put in, the function could give out cat: it doesn't matter,0707

as long as the first thing, the first set we are coming from, can't split as it comes out.0712

We can land on the same place, but we can't be coming from the same place and go to two different locations.0719

We always have to follow one rule; because we are following one rule, we can't land on two different things.0725

Let's look at a non-numerical example: before we start telling you about how functions work on numbers,0731

let's consider an example of one that works on something totally not about numbers.0735

Let's think about a function that gives initials: we will define...f is going from names spelled with the Roman alphabet0740

(names like Vincent or John, not things that are spelled with characters that we can't express in the Roman alphabet),0747

and it is going to go to letters from the Roman alphabet.0755

So, f(x) equals the first letter of x; now, if we say, "Wait, we know that the first letter of x is x!"--0759

yes, but what we are talking about is names: x is a placeholder, remember?0768

We talked about variables: the idea of a variable is that it is a placeholder.0773

So, x is just sort of keeping the spot warm, until later, when we put in the name.0777

So, if we decide to put Vincent into the function, then this x on the left side tells us where to put Vincent on the right side.0782

So, Vincent will come in here on the right side, as well.0791

We will have Vincent go on the left, and Vincent will go on the right.0795

f(Vincent) would be V: we cut it off just to the first letter.0799

f(Nicole) would put out N; f(Padma) would give out P; f(Victor) would give out V; f(Takashi) would give out T.0804

Whatever we put in, it will give out just that single letter.0811

So, if we were to turn this into a diagram, we could have Vincent here, Nicole next to Vincent, Padma, Victor, and then finally Takashi.0815

And so, this is where we are coming from; and then, we are going to letters.0832

So, we have V and N and P and T...and let's put in another letter, like...say S and Q.0842

Vincent gets mapped to V; Nicole, by this function, gets mapped; Padma gets mapped to P; Victor also gets mapped to V.0853

Takashi gets mapped to T; but do S and Q get used? Not for this set of names.0863

Maybe if we put in Susan, or we put in...there has to be a name with Q that I don't know...0868

let's pretend that the name is simply Queen...I am sure that there is a name...a really weird spelling of the name Cory?...0876

there is a name out there that is spelled with a Q; I just don't know it immediately.0883

So, there is something out there that can fill up that S, and that can fill up that Q; we just don't have it in what we are looking at so far.0886

So, there might be other things that we are not hitting on the right;0893

but everything that we have on the left is what is getting mapped to things on the right.0896

So, the functions we use...of course, it is no surprise; this is math--we are probably going to be talking about numbers.0901

So, it shouldn't come as a surprise; we are going to concentrate on using these functions with numbers.0907

Functions, as we just saw, can be used for lots of things; but we will focus on functions and the real numbers.0911

Unless we are told otherwise, we will assume that every function takes in real numbers and outputs real numbers.0917

That is to say, f is taking in reals and then giving out reals.0922

OK, so when we are given a function, we will usually be told what its rule is--how it maps inputs to outputs.0927

So, for example, if f(x) = x2 + 3, its rule is "Square the input," since x is our input;0934

then, what we do is...we first square the input, and then we add 3; square the input and then add 3.0941

That is its rule; that is how it works.0946

Notice that x acts as a placeholder; just like it did with the names, it acts as a placeholder.0948

It is not that x is really the thing we are worried about being acted on.0953

It is just telling us what is going to happen to whatever we plug into this function.0956

If we plug in 3, what will happen to 3?0960

If we plug in 50, what will happen to 50?0962

If we plug in smiley-face, what will happen to smiley-face?0964

x is just there to sort of keep a spot warm: it is telling us, "Here is the place; things will go into this place."0967

And things will go into this place, wherever I show up on the right side, as well.0974

If we want to use a function, if we want to evaluate a function at a specific value, we just apply this rule to whatever our input value is.0979

In practice, this turns out to actually be really simple.0986

Usually, we are given a formula for each function; so we just follow the method of substitution.0988

Remember, we take whatever we are substituting in; we wrap it in parentheses; and then we see what we get.0992

For example, our function is f(x) = x2 + 3; then, to find f(7), we just plug in.0997

7 is what we are plugging in; so we have 7 in this spot, and a 7 will go in here.1004

We wrap that in parentheses, just in case; in this case, we don't have to, but we will see why it is useful to always remember to wrap it in parentheses.1009

72 + 3...72 is 49; 49 + 3...we get 52.1016

If we want to look at a slightly more complex example, though, we see why it is so important to wrap your substitutions in parentheses.1021

If we consider a slightly more complex input, like a + 7, then we have to have it in parentheses,1029

because it is not just the a that gets squared; it is not just the 7 that gets squared; it is all of that thing that went in.1034

All of that thing is both the a and the + 7; it is (a + 7); it is that whole number combined.1041

It is not a2 + 7; it is not a + 72; it is (a + 7), the whole thing squared; and then, plus 3.1047

A good way to see the behavior of a function is by creating a table of values; sometimes we call it a T-table, because it has the shape of a T.1057

On one side, we have input values, while the other side shows us what the function outputs when given that input.1065

So normally, the left side will be our input value, and the right side will be our output value.1071

So, for example, if f(x) = x2 + 3, then we can give out a bunch of values for it.1076

So, if we want to figure out what happens to f(-2), we just follow the normal thing.1080

f(-2), so we plug it in...(-2)2 + 3...we get 4 + 3; we get 7, and that 7 shows up here.1086

If we want to figure out what f(-1) is, we do the exact same thing: (-1)2 + 3, 1 + 3, and 4.1095

And that 4 shows up here; and so on, and so forth.1103

We just plug in, based on this rule...whatever the rule we have been given...we plug in whatever our input is,1106

whatever the thing on the left is, any of these numbers.1112

And then, once we figure out what this number is here, we figure out, we evaluate, and we get what its corresponding value is on the right side.1116

And we write that in, and that is how we make a table of values.1122

Having this table is often a very useful way to quickly analyze and see what is happening in a function over a large range of possible inputs.1126

Domain: the domain is the set of all inputs that the function can accept.1135

The domain is what can go into the function: it is the inputs that we are allowed to use.1141

It is what our machine can eat without breaking down.1146

Well, we generally assume that all of ℝ can be used as inputs--all of the real numbers can be used as inputs.1150

Sometimes, certain values will break our function; the output won't be able to be defined.1154

Thus, our domain is normally going to be all of the real numbers, except those numbers that break our function.1160

Occasionally, we might actually get things where we are going to be given an explicit domain--1165

like just evaluate it from -3 to 3--and forget everything beyond those -3 and 3 values.1169

But normally, we are going to assume all of ℝ, except those things that break our function.1175

Let's see an example: if we had f(x) = 1/x, the function would be defined, as long as we don't divide by 0.1179

If we have x = 0, though, then f(0) gets us 1/0.1188

Are we allowed to do that? No--that is very bad; we cannot divide by 0.1196

So, we are not defined there; everything else works, though.1202

If we plug in anything that isn't a 0, it works out fine.1206

So, everything is defined, as long as x is not 0; so our domain is all numbers, except 0.1208

The domain of f, to show all numbers except 0, is everything from -∞ up to 0, not including the 0,1214

and then union with everything from 0, not including the 0, to ∞.1221

That is just another way of expressing all of the real numbers, with the exception of 0.1225

Now, for now, we mostly only have to watch out for dividing by 0 and taking square roots of negative numbers.1230

Those are the only two things we have to worry about breaking functions.1235

However, you can't take the square root of a negative number, because what could you square that would still have a negative with it?1237

Any number, squared, becomes positive; so we can't have the square root of a negative number,1243

because it would be impossible to give me a number that you could square1247

into making it negative--at least as far as the real numbers are concerned.1250

Later on we will talk about the complex; but that is for later.1253

Right now, we only really have to worry about dividing by 0 and taking square roots of negative numbers.1257

Those are the things to watch for; that is where our domain will break down.1261

Later in the course, we will have a little bit more to worry about; we will also have to worry about inverse trigonometric functions.1264

Those are only defined over certain things; and also logarithms have some parts that they are not allowed to take, either.1269

But right now, it is just dividing by 0 and taking square roots of negative numbers.1274

And later on, much later in the course, after we see these ideas, we will have to think about them, as well,1277

when we are thinking about the idea of what can go into a function.1281

Domain is what goes in; range is what comes out.1284

Range is the set of all possible outputs a function can assign, given some domain.1289

With some domain to start with, these values are what is able to come out: the range is what can come out, given some domain.1294

These values will always be in the real numbers, unless we are dealing with a set that isn't working in the reals.1302

But they don't necessarily cover all of the reals.1307

For example, that function we were working with before, f(x) = x2 + 3:1309

the lowest value that f can output is 3, because the smallest number we can make with x2...1313

well, x2 always has to be greater than or equal to 0, because there is no number1319

that we can plug into x and square that will cause it to become negative.1323

The lowest we can get that down to is a 0, so the lowest we can make this whole thing is when this is a 0, plus 3; so the lowest possible output is 3.1327

We can produce any value above 3 with x2, though, so we can just keep going up and up.1335

So, our range would be everything from 3, including 3, up until infinity.1340

So, it is all of the reals from 3, including 3, and higher; great.1344

If we want to look at an example that doesn't use numbers, we could talk about that initial function,1349

that function that ate names and gave out first initials, from earlier in this lesson.1353

In that case, if the domain is all names, then the range is all 26 letters of the Roman alphabet,1357

even though I still can't think of any names that start with a Q...Queen...let's say Queen counts.1362

OK, Queen Latifah, right?--it has to count; then, we can have that be the range--26 letters for the Roman alphabet.1372

So, because if we are looking at all the names that could possibly exist...1379

well, there is Albert; there is Bill; there is Charles; there is Doug; there is Elizabeth...and so on, and so on, and so forth.1383

So, there is always something that will put that out.1391

But if we restricted the domain to the five names that we saw earlier, Vincent, Nicole, Padma, Victor, and Takashi,1394

then we only had four letters show up--we just had N, P, T, and V show up.1400

So, in that case, if we restricted our domain to a smaller thing, our range would also shrink.1405

So, the range depends on what our domain is.1410

If we are looking at...normally we look at everything that can go into the function, and that is normally how we think of the domain.1412

So, the range is everything that could come out.1418

But sometimes, we will be given a more restricted domain, and we have to think in terms of that more restricted domain.1420

All right, we are ready for some examples.1425

First, there are nice, easy ones to get us warmed up to this idea of plugging in.1427

If f(x) = 3x - 7, what is f(2)?1430

We just plug in...if we use red for this...f(2)...we plug in 3; plug in that 2; minus 7...3 times 2 equals 6; 6 minus 7...so we get -1.1434

Let's use blue for this one: if we have f(-4), then 3...we plug in that -4, minus 7.1447

3 times -4 is -12; minus 7...we get -19.1455

Oh, no! What if we have to use something that is variable? No problem.1460

We still just follow the exact same rules: f(a)...well, what happened to x?1465

It became 3x - 7, so now it is going to become 3a - 7, so we get 3a - 7.1470

And what if we want to do b + 8? The same thing--f(b + 8) = 3(b + 8) - 7.1479

So, we have to distribute; and notice how important it was that we put it in parentheses.1490

If we had just plugged in this 3b + 8, that would be totally different than 3(b + 8).1495

And that is what it really has to be, because it is everything in here that got plugged in, not just the b.1501

The b and the 8 don't get to be separated now; they have to go in together.1506

So, 3(b + 8) - 7...we would get 3b + 24 - 7, which is equal to 3b + 17.1509

The next one: what if we wanted to fill in a table, g(z) = z2 - 2z + 3?1522

If we had to fill in this table, then we could do g(-1), (-1)2 - 2(-1) + 3,1527

equals 1 (-1 times -1 is 1), minus 2(-1), so plus 2, plus 3, equals 6; so we get 6 here.1539

Next, g(0): 02 - 2(0) + 3...that simplifies to just 3, because of the 0's; they disappear.1551

If we want to plug in g(1), then we get 12 - 2(1) + 3, so 1 - 2 + 3 comes out to 2.1564

We plug in g(2); we get 22 - 2(2) + 3 = 4 - 4 + 3, which is 3.1581

We plug in 10; we get 102 - 2(10) + 3; 102 is 100, minus 2 times 10 (is 20), plus 3 equals 83.1600

There we go: so you just plug into the function exactly as you would to set up this table.1621

You are told what your input is; and then, over on the right is your output, based on the rules of the function.1625

The function gives us some rules, and so we plug in inputs like -1, and -1 goes through: (-1)2 - 2(-1) + 3; we get 6.1634

And that is what is going on when we are making a table of values.1645

If h(x) = 2x2 + bx + 3, and we know that h(3) = 15, what is b?1649

So in this case, we are looking to figure out what b is.1655

Now, we know that h(3) = 15; so we need to somehow use this to figure out b.1660

So, we think, "I could plug in 3, and I would get something different than just 15."1667

So, h(3), based on the rule, is 2(32)...we are switching for where all of the x's show up;1672

x here; x here; that is it; so 2(32) + b(3) + 3...so we get 2(9) + 3b + 3, which is 18 + 3b + 3, or 3b + 21.1681

Now, at this point, we say, "Right; I also know that h(3) is 15; well, this is still h(3), right?"1707

So now, we put h(3) = 15, and we swap it out, and we get 15 must equal what we know h(3) is.1715

We know that h(3) is equal to 3b + 21; and we also know that h(3) is equal to 15.1725

So, since h(3) is two different things, but it is still just h(3), we know that they must be the same thing; otherwise there is no logic there, right?1730

So, 15 = 3b + 21; we subtract 21 from both sides; we get -6 = 3b; we divide by 3 on both sides, and we get -2 = b.1739

The next one: What is the domain and range of f(x) = 12 - √(x + 3)?1763

Now, remember: domain is what can go in; range is what can come out.1771

What we first one to do is figure out the domain first: what can go in without breaking this function?1794

So, is there anything that can break in this function?1801

We say, "Oh, right, the square root breaks when there is a negative inside."1803

We can't take the square root of -1, because there is no number that you can give me--1812

at least no real number that you can give me--that would square to give us -1.1816

You give me any positive number; it comes up positive; you give me any negative number; it comes out positive.1822

You give me 0; it comes out 0; so there is no number you can give me that will give out a negative number when squared.1825

So, square root breaks when we are trying to put a negative inside of it.1831

So, when will this break? √(x + 3) breaks when we have a negative inside.1835

So, when is (x + 3) going to be negative? when x is less than -3.1842

So, if x is less than -3, if x is more negative than -3, then this will be a negative value inside.1850

If, for example, we use -4, then we will get the square root of -1.1860

If we put in negative fifty billion, then we will get the square root of negative fifty billion plus 3, which would definitely still be negative.1863

So, it only stops being a negative inside when we actually get to -3.1869

-3 is an allowed value, because √(-3 + 3) would be √0; we do know the square root of 0--it is 0.1873

So, the domain works for -3 and higher; everything is still reasonable higher than that.1880

Our domain is going to include -3, and it is going to go for anything higher than that.1888

So, that is our domain; if we want to figure out what the range is, then the question is, "What can f(x) put out?"1895

So, notice: we have 12 - something; that something, √(x + 3)...square root can give out any number.1907

If you put in √0, √1, √4, √9...you are going 0, 1, 2, 3, and you can make any number in between that.1920

12 - something...what is the smallest that something could be?1928

The smallest number that that something could be is 0; so that is smallest when √0 = 0.1932

The biggest number we can get is 12; 12 is the highest number we can get, the largest number we can get out of this function.1946

What is the smallest number we can get? Well, you can just keep giving me larger and larger x1956

to make our square root a bigger and bigger negative number, on the whole.1961

It would be minus larger and larger numbers; so 12 minus larger and larger numbers...we can keep going down.1965

So, any number below 12 can be achieved, because we can just keep having the square root give out slightly larger and slightly larger numbers, which...1971

Since we are subtracting by these larger and larger numbers, we will keep going down.1982

So, any number below 12 can be achieved; so we have our range--it is going to be everything from the lowest possible,1986

all the way, anywhere up from negative infinity, up until 12.1997

Now, we ask ourselves, "Can we actually achieve 12?" Yes, we can.2002

We can actually get to 12, so we include 12; so our range is from negative infinity to 12--there is our answer.2006

The final one: we have a word problem: Give the area of a square, A, as a function of the square's perimeter, p.2013

And then, also say what is the domain of the area as a function of the perimeter.2020

First, as we talked about in the word problems, let's set up what our variables are.2025

Nicely, this problem already gave us our variables; but we will just remind ourselves: A is the area of the square, and p is the perimeter of the square.2030

So, it also probably wouldn't hurt to draw a picture, so we could see what is going on a little more easily.2051

We have a square here; here is our square, and we are talking about the area of it and the perimeter of it.2056

So, that is everything that we don't immediately know: we don't know the area; we don't know the perimeter.2065

They are going to be somehow connected, because we somehow want to be able to make a function out of area,2069

where we plug in the perimeter, and it gives out an area.2073

We basically want an equation that has area on the left, and then things involving perimeter.2076

We are solving for area in terms of perimeter; that is another way of looking at what this function is going to be.2081

We need some way to be able to connect these two ideas: how can we connect the area of a square to its perimeter?2086

Well, maybe we don't see a way right away; but let's just think, "Well, how do you find the area of a square?"2092

Well, it is its side times its side, its side squared; so the area of a square...2099

Now, we might as well go back, and we will set up a new variable--we didn't have that before--side of square.2105

A side of the square is a way to get our area; so area equals side squared.2114

Now, we still want some way to connect the area to the perimeter.2120

So, what we want is...well, we might not be able to correct them directly, but we have area connected to sides.2123

Maybe we can connect perimeter to side...oh, right, yes...if you have forgotten what a perimeter is, what do you do?2129

You just go and look it up: you have access to all sorts of information at your fingertips--it is so easy.2135

If you look up perimeter, thinking, "Oh, I have heard this before; I can't remember what it is,"2140

type it into an Internet search; the next thing you know, you will have a definition for what perimeter is.2144

So, perimeter is all of the sides added together; we have four sides, so perimeter is equal to side + side + side + side--2148

all of the sides of our square, or 4s.2160

So now, we have a way of being able to have area talk to perimeter.2164

Area equals side squared; perimeter equals 4 times side; so perimeter/4 equals side.2168

Now, we can take this, and we can plug it in here.2178

Area equals...since we are plugging and we are substituting, we do it with parentheses...squared.2183

Area equals perimeter squared, over 16 (we have to square the top and the bottom).2190

And there we are--we can think of area equals perimeter squared over 16 as a function,2196

because it only depends on what we plug in for perimeter.2201

Area will vary as we put in different things; so we can think of it as a function acting on perimeter.2205

We plug in the number from perimeter, and it gives out what the area has to be.2210

So, we can just rewrite this as: area is a function of perimeter, or it is equal to the perimeter squared, over 16.2214

It is just a different way of thinking about it: we can think of it as an equation, or we can think of it as a function2223

where it just works the exact same way that the equation worked.2229

There is no functional difference between area of perimeter equals perimeter squared over 16--2233

the area based on a function using our perimeter equals p2/16--compared to area equals p2/16.2238

They have the same effect; it is just two slightly different ways of talking about it.2245

But in either case, it is plugging in a number for perimeter, then figuring out what the area has to be.2249

That is a function for area as a function of a square's perimeter.2255

Now, how can we get its domain? We talked about, before: the domain is everything that we can plug in without breaking it.2260

Now, that is mostly true; but there is one little thing here.2269

The domain also has to make sense; we can't break the world.2272

We wouldn't break this function...we could plug anything we want into this function.2278

You plug in any real number in for p, and it would make sense; we would get a number out of it.2284

You can plug in 50; you can plug in 0; you could plug in -10; it makes sense--we would get a number out of it.2289

But the domain has to make sense; it doesn't have to just make sense in our function--it has to make sense in how we have thought about the function.2294

How did we think about the function? It is a square, right?2307

It is a real object--it is a thing; we could talk about its shape and how its dimensions are.2311

Would it make sense for it to have a perimeter of a negative? No, because it doesn't make sense for the sides to be negative.2317

Would it make sense for the perimeter to be 0? No, because then it would just be a speck--it wouldn't be a square.2324

There would be no area possible to be contained inside, because we would have no side lengths if we had a perimeter of 0.2329

So, it must be the case that our p, perimeter, is allowed to vary only from 0 up to infinity,2335

because it can't have a domain below 0, and it can't have a domain of 0,2344

because while it doesn't break our function itself, it breaks the idea of what the function means.2348

It is meaningless to talk about plugging in a perimeter that is negative or a perimeter that is 0,2354

because then it is not the perimeter of anything; we don't actually have a shape there.2359

We have to be having our domain make sense, as well.2362

If we just have a function, it can't break the function.2365

But if we have the function in the context of a word problem, it also has to make sense with everything else happening in the word problem.2368

All right, I hope that all made sense, because that just laid an important groundwork.2374

You are going to need to know this for the rest of your time in math.2378

So, it is really great that we got this covered here.2381

Having a really strong understanding of what it means for something to be a function2382

is going to help you out in so many different places in math.2386

It is going to help you with all sorts of things--it is really great that we covered that here.2388

All right, see you at Educator.com later--goodbye!2391