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INSTRUCTORSCarleen EatonGrant Fraser
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Lecture Comments (6)

0 answers

Post by Munqiz Minhas on August 25 at 06:49:04 PM

Do you have any videos on Interval Notations and Sets?

1 answer

Last reply by: Dr Carleen Eaton
Mon Apr 9, 2012 6:32 PM

Post by jeremiah dulla on April 7, 2012

Do you have any videos on graphing Functions and like Transformations of Functions Horizontal Compression,expansion etc.

0 answers

Post by Jasmine Valdovinos on August 18, 2011

why did you place a 3 on the domain shouldnt it be -1,0,1,2...

1 answer

Last reply by: Dr Carleen Eaton
Sat May 28, 2011 11:09 PM

Post by Victoria Jobst on May 28, 2011

Is example 2 discrete or continuous?

Relations and Functions

  • Each relation or function has a domain and a range.
  • Functions may be one to one, and may be discrete or continuous.
  • Functions satisfy the vertical line test: any vertical line crosses the graph at most once.
  • One variable, usually x, is the independent variable. The other variable is the dependent variable.

Relations and Functions

Given the relation R = { (5, − 2),(1,2),(3,2),(1,1),(2, − 1)} . Give the Domain and Range. Is R a function?
  • Domain = all x's taken up by the relation
  • Range = all the y's taken up by the relation
  • Domain = { 1,2,3,5}
  • Range = { − 2, − 1,1,2}
  • Create a Map to check if the relation is a Function.
  • Domain 1 2 3 5
    Maps To 1 & 2 -2 2 -2
The relation R is not a function. There are two arrows coming out from the same x = 1
Given the relation R = { (2, − 3),(4,0),(3,2),(5,1),(2,2)} . Give the Domain and Range. Is R a function?
  • Domain = all x's taken up by the relation
  • Range = all the y's taken up by the relation
  • Domain = { 2,3,4,5}
  • Range = { − 3,0,1,2}
  • Create a Map to check if the relation is a Function.
  • Domain 2 3 4 5
    Maps To -3 & 2 2 0 1
The relation R is not a function. There are two arrows coming out from the same x = 2
Given the relation R = { (1, − 1),(3,1),(4,2),(5,0),(6,2)} . Give the Domain and Range. Is R a function?
  • Domain = all x's taken up by the relation
  • Range = all the y's taken up by the relation
  • Domain = { 1,3,4,5,6}
  • Range = { − 1,0,1,2}
  • Create a Map to check if the relation is a Function.
  • Domain 1 3 4 5 6
    Maps To -2 1 2 0 2
The relation R is a function. There are no two arrows coming out from the same value of x.
The relation R is given by the equation y = 2x + 1. Is R a function? What is the domain and range?
Is R discrete or continuous?
  • Create a table of values to check if the relation is a function.
  • x y=2x+1
    -2 y=2(-2)+1=-3
    -1 y=2(-1)+1=-1
    0 y=2(0)+1=1
    1 y=2(1)+1=3
  • The relation is a function because it is 1 - to − 1
  • Find the Domain
  • Find the Range
  • Domain = All real values
  • Range = All real values
The relation R is continous becuase it represents a linear equation.
Graph the relation R given by 4x − 2y = 8. Is R a function? What is the domain and range?
Is R discrete or continuous?
  • x 4x-2y=8
    -1 4(-1)-2y=8
      -2y=12
      y=-6
    0 4(0)-2y=8
      -2y=8
      y=-4
    2 4(2)-2y=8
      8-2y=8
      -2y=0
      y=0
    3 4(3)-2y=8
      12-2y=8
      -2y=-4
      y=2
  • Find the Domain
  • Find the Range
  • Domain = All real values
  • Range = All real values
The relation R is continous becuase it represents a linear equation.
Let f(x) = 3x2 + 5x find f( − 1),f(2k)
  • To evaluate the given function at the given values, substitute the input where ever there is an x.
  • f( − 1) = 3( − 1)2 + 5( − 1)
  • f( − 1) = 3(1) − 5
  • f( − 1) = − 2
  • f(2k) = 3(2k)2 + 5(2k)
  • f(2k) = 3(4k2) + 10k
  • f(2k) = 12k2 + 10k
f( − 1) = − 2
f(2k) = 12k2 + 10k
Let f(x) = x3 + x2 find f( − 1),f(3k)
  • To evaluate the given function at the given values, substitute the input where ever there is an x.
  • f( − 1) = ( − 1)3 + ( − 1)2
  • f( − 1) = ( − 1)( − 1)( − 1) + ( − 1)( − 1)
  • f( − 1) = − 1 + 1 = 0
  • f(3k) = (3k)3 + (3k)2
  • f(3k) = (3k)(3k)(3k) + (3k)(3k)
  • f(3k) = 27k3 + 9k2
f( − 1) = 0
f(3k) = 27k3 + 9k2
Let f(x) = − x3 − x2 find f( − 1),f(2)
  • To evaluate the given function at the given values, substitute the input where ever there is an x.
  • f( − 1) = − ( − 1)3 − ( − 1)2
  • f( − 1) = − ( − 1)( − 1)( − 1) − ( − 1)( − 1)
  • f( − 1) = − ( − 1) − (1) = 0
  • f(2) = − (2)3 − (2)2
  • f(2) = − (2)(2)(2) − (2)(2)
  • f(2) = − (8) − (4) = − 12
f( − 1) = 0
f(2) = − 12
Let f(x) = x4 + x2 find f( − 1),f( − k)
  • To evaluate the given function at the given values, substitute the input where ever there is an x.
  • f( − 1) = ( − 1)4 + ( − 1)2
  • f( − 1) = ( − 1)( − 1)( − 1)( − 1) + ( − 1)( − 1)
  • f( − 1) = (1) + (1) = 2
  • f( − k) = ( − k)4 + ( − k)2
  • f( − k) = ( − k)( − k)( − k)( − k) + ( − k)( − k)
  • f( − k) = k4 + k2
f( − 1) = 2
f( − k) = k4 + k2
Let f(x) = x5 + x3 find f( − 1),f( − k)
  • To evaluate the given function at the given values, substitute the input where ever there is an x.
  • f( − 1) = ( − 1)5 + ( − 1)3
  • f( − 1) = ( − 1)( − 1)( − 1)( − 1)( − 1) + ( − 1)( − 1)( − 1)
  • f( − 1) = ( − 1) + ( − 1) = − 2
  • f( − k) = ( − k)5 + ( − k)3
  • f( − k) = ( − k)( − k)( − k)( − k)( − k) + ( − k)( − k)( − k)
  • f( − k) = − k5 − k3
f( − 1) = − 2
f( − k) = − k5 − k3
f(x) = x5 + x4 + x3 find f( − 1),f( − 2k)
  • To evaluate the given function at the given values, substitute the input where ever there is an x.
  • f( − 1) = ( − 1)5 + ( − 1)4 + ( − 1)3
  • f( − 1) = ( − 1)( − 1)( − 1)( − 1)( − 1) + ( − 1)( − 1)( − 1)( − 1) + ( − 1)( − 1)( − 1)
  • f( − 1) = ( − 1) + (1) + ( − 1) = − 1
  • f( − k) = ( − k)5 + ( − k)4 + ( − k)3
  • f( − k) = ( − k)( − k)( − k)( − k)( − k) + ( − k)( − k)( − k)( − k) + ( − k)( − k)( − k)
  • f( − k) = − k5 + k4 − k3
f( − 1) = − 1
f( − k) = − k5 + k4 − k3

*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.

Answer

Relations and Functions

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
  • Coordinate Plane 0:20
    • X-Coordinate and Y-Coordinate
    • Example: Coordinate Pairs
    • Quadrants
  • Relations 2:14
    • Domain and Range
    • Set of Ordered Pairs
    • As a Table
  • Functions 4:21
    • One Element in Range
    • Example: Mapping
    • Example: Table and Map
  • One-to-One Functions 8:01
    • Example: One-to-One
    • Example: Not One-to-One
  • Graphs of Relations 11:01
    • Discrete and Continuous
    • Example: Discrete
    • Example: Continous
  • Vertical Line Test 14:09
    • Example: S Curve
    • Example: Function
  • Equations, Relations, and Functions 17:03
    • Independent Variable and Dependent Variable
  • Function Notation 19:11
    • Example: Function Notation
  • Example 1: Domain and Range 20:51
  • Example 2: Discrete or Continous 23:03
  • Example 3: Discrete or Continous 25:53
  • Example 4: Function Notation 30:05

Transcription: Relations and Functions

Welcome to Educator.com.0000

For today's Algebra II lesson, we are going to be discussing relations and functions.0002

And recall that some of these concepts were discussed in Algebra I, so this is a review.0008

And if you need a more detailed review, check out the Algebra I lectures here at Educator.0013

Beginning with the concept of the coordinate plane: the coordinate plane describes each point as an ordered pair of numbers (x,y).0021

The first number is the x-coordinate, and the second is the y-coordinate.0031

For example, consider the ordered pair (-4,-2): this is describing a point on the coordinate plane with an x-coordinate of -4 and a y-coordinate of -2.0036

Or the pair (0,2): the x-value would be 0, and the y-value would be 2; this is the point (0,2) on the coordinate plane.0057

Or (3,5): x is 3; y is 5.0073

Also, recall that the quadrants are labeled with the Roman numerals: I, and then (going counterclockwise) quadrant II, quadrant III, and quadrant IV.0080

In the coordinate pairs in the first quadrant, the x is positive, as is the y.0097

In the second quadrant, you will have a negative value for x and a positive value for y, such as (-2,4)--that would be an example.0103

In the third quadrant, both x and y are negative; and then, in the fourth quadrant, x is positive; y is negative.0114

And we will be using the coordinate plane frequently in these lessons, in order to graph various equations.0127

Recall that a relation is a set of ordered pairs.0135

The domain of the relation is the set of all the first coordinates, and the range is the set of all the second coordinates.0140

A relation is often written as a set of ordered pairs, using braces to denote that this is a set,0149

and then the ordered pairs, each in parentheses, separated by a comma.0158

Sometimes, the relation is represented as a table; so, -2, 1; -1, 0; 0, 1; and 1, 2.0172

We will be doing some graphing of relations also, in just a little bit.0192

So, as discussed up here, the domain is the set of all first coordinates.0196

And the set of all first coordinates here would be {-2, -1, 0, 1}.0207

The range is the set of the second coordinates; so the range right here--all the y-values--is {1, 0, 1, 2}.0219

However, you don't actually need to write the 1 twice; so in actuality, it would be written as such.0232

It is OK to repeat the values if you want, but usually, we just write each value in the domain or range once; each is represented once.0245

Functions are a certain type of relation: so, all functions are relations, but not all relations are functions.0261

A function is a relation in which each element of the domain is paired with exactly one element of the range.0272

For example, consider the relation shown: {(1,4), (2,5), (3,8), (4,10)}.0280

Each member of the domain corresponds to exactly one element of the range.0298

We don't have a situation where it is saying {(1,4), (1,6), (1,8)}, where that member of the domain is paired with multiple members of the range.0305

Another way, again, to represent this as a table--another method that can be used--is mapping.0314

And mapping is a visual device that can help you to determine if you have a function or not,0321

by showing how each element of the domain is paired with an element of the range.0327

A map would look something like this: over here, I am going to put the elements of the domain, 1, 2, 3, and 8;0333

over here, the elements of the range: 4, 5, 8, and 10.0347

And then, using arrows, I am going to show the relationship between the two.0355

So, 1 corresponds to 4 (or is paired with 4); 2 to 5; 3 to 8; and (this should actually be 4) 4 to 10.0360

OK, so as you can see, there is only one arrow going from each element of the domain to each element of the range.0374

And that tells me that I do have a function.0383

Let's look at a different situation, using a table form: let's look at a second relation.0386

In this one, I am going to have (-2,2), (-3,2), (-4,5), and (-6,7).0392

And I am going to go ahead and map this: -2, -3, -4, and -6: these are my elements of the domain.0403

For the range, I don't have to write 2 twice; I am just going to write it once; 5, and 7.0416

OK, -2 corresponds to 2; -3 also corresponds to 2; -4 corresponds to 5; and -6 corresponds to 7.0422

This is also a function, so both of these are relations, and they are also functions.0436

It is OK for two elements of the domain to be paired with the same element of the range; this is allowed.0445

What is not allowed is if I were to have a situation where I had {1, 2, 3}, {4, 5, 6}; and I had 1 paired with 4, and 1 paired with 5.0452

So, if you have two arrows coming off an element of the domain, then this is not a function.0468

Here are two examples of relations that are also functions.0476

There is a specific type of function that is called a one-to-one function.0482

And a function is one-to-one if distinct elements of the domain are paired with distinct elements of the range.0486

In the previous example, we saw a situation where we did have a one-to-one function, and another situation where we did not.0494

OK, so to review: the ordered pairs in that first function that we just discussed were (1,4), (2,5), (3,8), and (4,10).0501

OK, and we can use mapping, again, to determine what the situation is with this relation (which is also a function).0519

The domain is {1, 2, 3, 4}; and the range is {4, 5, 8, 10}.0529

When I put my arrows to show this relationship, you see that distinct elements of the domain are paired with distinct elements in the range.0540

1 is paired with 4; they are each unique--each pair is unique.0552

Looking at the other function that we discussed: the pairs are (-2,3), (-3,2), (-4,3)...slightly different, but the same general concept...slightly different, though.0558

OK, here I have -2, -3, -4, and -6; over here, in the range, I have 3, 2...I am not going to repeat the 3--I already have that...and then 7.0587

-2 corresponds to 3; -3 corresponds to 2; -4 also corresponds to 3; -6 corresponds to 7.0605

This is still a function; OK, so these are both functions: function, function.0620

However, this is a one-to-one function; this is not one-to-one.0628

They are both functions, since each element of the domain is paired only with one element of the range.0639

But in this case, it is not a unique element of the range: these two, -2 and -4, actually share an element of the range.0648

In other words, this is unique; it is a one-to-one correspondence.0656

OK, we can graph relations and functions by plotting the ordered pairs as points in the coordinate plane, as discussed a little while ago.0662

There are a couple of types of graphs that you can end up with.0673

The first is discreet, and the second is continuous; let's look at those two different types.0676

Consider this relation: OK, so if I am asked to graph this relation, I am going to graph each point:0682

(-4,-2): that is going to be right here; (-2,1)--right here; here, (0,2), 0 on the x, 2 on the y.0698

This is a discrete function--discrete graph--discrete relation.0714

This actually is both a relation and a function; so it is a discrete relation or a discrete function.0721

And the reason is because I have a set of discrete points; they are not connected.0727

And I can't connect them, because I haven't been given anything in between, or a way to know if or what lies in between these.0734

I can't just connect them when I don't know; there could be a point up here, or actually this is just the entire relation.0741

So, I can just work with what is given.0747

OK, a different scenario would be if I am given a relation y=x+1.0749

And I can go ahead and plot this out, if I say, "OK, when x is -1, -1+1 is 0; when x is 0, y is 1; when x is 1, 1+1 is 2; when x is 2, y is 3."0759

OK, so I am going to go ahead and plot this out.0779

When x is -1, y is 0; when x is 0, y is 1; when x is 1, y is 2.0781

Let's remove this out of the way.0790

When x is 1, y is 2; when x is 2, y is 3.0793

Now, I have a set of points, because these are the points I chose.0798

But because I am given this equation, there is an infinite number of points in between.0801

I could have chosen an x of .5 to get the value 1.5 here, to fill that in--and on and on, until this becomes continuous and forms a line.0807

So, this is a continuous function: the graph is a connected set of points,0821

so the relation (or the function in this case, since we do have a function) is continuous relation or continuous function,0834

because I have a line; whereas this, which is a set of points, is a discrete relation.0841

OK, one visual way to tell if a relation is a function is using the vertical line test.0849

And a relation is a function if and only if no vertical line intersects its graph at more than one point.0856

This is most easily understood through just working through an example.0866

Consider if you were given the following graph.0870

OK, the vertical line test: what you are seeing is, "Can you put a vertical line0874

somewhere on the graph so that it intersects the graph at more than one point?"0880

And I can: I put a vertical line here, and it intersects this graph at 1, 2, 3 places.0887

Over here, it only intersects at one place; that is fine; but if I can draw a vertical line anywhere on the graph0897

that intersects at more than one place, then we say that this failed the vertical line test.0903

And when something fails the vertical line test, it means that it is not a function.0912

The reason this works is that, if two or three or more points share the same x-value,0919

then they are going to lie directly above or below each other on the coordinate plane.0926

For example, looking right here, I have x = 3; x is 3; y is 0.0931

Then, I look right above it, up here: again, x is 3, and y is...say 2.1--pretty close.0940

Then, I look up here; again, x is 3, and y is about 4.6, approximately.0950

So, when x-values are the same, but then the y-values are different, that is telling me0959

that members of the domain are paired with more than one member of the range; by definition, that is not a function.0968

OK, consider a different graph--consider a graph like this of a line--a straight line.0975

OK, now, anywhere that I pass a vertical line through--anywhere on this graph--it is only going to intersect at one point.0984

So, this passed the vertical line test.0996

Therefore, this line, this graph, represents a function.1006

OK, so the vertical line test is a visual way of determining if a relation is a function.1014

Working with equations: an equation can represent either a relation or a function.1024

If an equation represents a function, then there is some terminology we use.1031

And let's start out by just looking at an equation that represents a function.1036

The variable corresponding to the domain is called the independent variable, and the other variable is the dependent variable.1043

So, here I have x and y; and let's look at some values--let's let x be -1.1050

Well, -1 times 2 is -2, minus 1--that is going to give me -3.1057

When x is 0, 0 times 2 is 0, minus 1 is -1.1063

When x is 1, 1 times 2 is 2, minus 1--y is 1.1071

When x is 2, 2 times 2 is 4, minus 1 gives me 3.1077

So, looking at how this worked, x is the independent variable.1084

The value of x is independent of y; I am just picking x's, and here it could be any real number.1099

We sometimes also say that this is the input; and the reason is that I pick a value for x (say 0),1109

and I put it in--I input it into the equation; then, I do my calculation, and out comes a y-value.1116

So, the value of y is dependent on x; therefore, it is the dependent variable; and we also sometimes say that it is the output.1127

You put x in and do the calculation; out comes the value of y; so x is independent, and y is dependent.1141

The notation that you will see frequently in algebra is function notation.1151

We have been writing functions like this: y = 4x + 3; but you will often see...1157

instead of an equation written like this, if it is a function, you will see it written as such.1165

And when we say this out loud, we pronounce it "f of x equals 4x plus 3."1170

And we are talking about the value of a function for a particular value of x.1179

So, we say, "The function of f at a particular x."1186

Let's let x equal 3; then, we can talk about f of 3--the value of the function, the value of y,1193

of the dependent variable, when the independent variable, x, is 3.1204

And in that case, since it is telling us that x is 3, I am going to substitute in 3 wherever there is an x.1209

And I could calculate that out to tell me that f(3) is...4 times 3 is 12, plus 3...so f(3) is 15.1217

Here, x is an element of the domain, of the independent variable; f(x) is an element of the range.1231

So again, we are going to be using this function notation throughout the remainder of the course.1246

Looking at the first example: the relation R is given by this set of coordinate pairs.1253

Give the domain and range, and determine of R is a function.1260

Well, recall that the domain is comprised of the first element of each of these coordinate pairs.1267

So in this case, the domain would be {1, 2, 6, 5, 7}.1275

The range: the range is comprised of the second element of each ordered pairs, so I have 4, 3...4 again;1291

I don't need to write that again; 3--I have 3 already; and 5; I am just writing down the unique elements.1303

This is the domain, and this is the range.1309

Now, is R a function? Well, I can always use mapping to just help me determine that.1311

And I am going to write down my members of the domain, and my elements of the range.1320

And then, I am going to use arrows to show the correspondence between each:1330

1 and 4--1 is paired with 4; 2 is paired with 3; 6 is also paired with 4; 5 is paired with 3; and 7 is paired with 5.1333

Now, I am looking, and I only see one arrow leading from each element of the domain.1350

There is no element of the domain that is paired with two elements of the range.1354

So, in this case, this is a function; so, is R a function? Yes, this relation is a function--R is a function.1359

So, always double-check and make sure you have answered each part.1373

I found the domain; I found the range; and I determined that R is a function.1375

OK, the relation R is given by the equation y=2x2+4; is R a function?1384

What are the domain and range? Is R discrete or continuous?1393

Let's just look at some values for x and y to help us determine if this relation is a function.1400

If I let x equal -1, -1 times -1 is 1, times 2 is 2, plus 4 is 6.1412

OK, if x is 0, this is 0, plus 4--that gives me 4.1422

If x is 3, 3 squared is 9, times 2 is 18, plus 4 is 22.1428

So, as you are going along, you can see that, for any value of x, there is only one value of y; therefore, R is a function.1434

What is the domain? Well, I could pick any real number for an x-value that I wanted, so the domain is all real numbers.1450

You might, at first glance, say, "Oh, the range is all real numbers, as well"; but that is not correct, because look at what happens.1467

Because this is x2, whenever I have a negative number, it becomes positive; if I have a positive number, it stays positive, of course.1475

Therefore, if I have, say, -1, that becomes 1; this becomes 6.1486

So, I am not going to get any value lower than...for y, the smallest value I will get is for when x is 0.1493

OK, so if x is 0, y is 4; because -1 is going to give me a bigger value--it is going to give me 6.1505

If I do -2, that is going to be 4 times 2 is 8, plus 4 is 12.1510

So, the lowest value that I will be able to get for y will occur when x is 0.1516

And that is going to give me a y-value of 4.1522

Therefore, the range is that y is greater than or equal to 4.1525

So, the most difficult part of this was just realizing that the range is not as broad as it looked initially.1530

Because this involves squaring a number, there is a limit on how low you are going to go with the y-value.1538

So, this is a range with a domain of all real numbers, and a range of greater than or equal to 4.1545

OK, in Example 3, graph the relation R given by 2x - 4y = 8.1559

Is R a function? Find its domain and range. Is R discrete or continuous?1572

OK, so graph the relation given by 2x - 4y = 8.1583

Let's go ahead and find some x and y values, so that we can graph this.1590

When x is 0, we need to be able to solve for y; when x is 0, let's figure out what y is.1598

0 - 4y equals 8; therefore, y equals -2 (dividing both sides by -4).1607

OK, when x is 2, 2 times 2 minus 4y equals 8; that is 4 minus 4y equals 8; that is -4y equals 4; y = -1.1615

And let's do one more: when x is -2, this is going to give me -4 - 4y = 8.1635

That is going to then give me, adding 4 to both sides, -4y = 12, or y = -3; that is good.1646

All right, so when x is 0, y is -2; when x is 2, y is -1; when x is -2, y is -3, right here.1657

I am asked to graph it; and I have some points here that I generated,1686

but I also realize that I could have picked points in between these, which would actually end up connecting this as a line.1691

So, I am not just given a set of ordered pairs; I am given an equation that could have an infinite number of values for x,1702

which would allow me to graph this as a continuous line.1707

Therefore, I graphed the relation...is R a function?1713

Is R discrete or continuous? Well, I have already answered that--seeing the graph of this, I know that this is continuous.1720

And let's see, the next step: is R a function?1731

Yes, it is a function, because if I look, for every value of x (for every value of the domain), there is one value only of the range.1746

So, every element of the domain is paired with only one element of the range.1758

It is continuous, and it is a function.1762

Find the domain and range: well, this is another case where I could choose x to be any real number, so it would be all real numbers--any real number.1765

Here, the situation is the same for the range--all real numbers.1778

Depending on my x-value, I could come up with infinite possibilities for what the range would be, what the y-value would be.1784

R is a function; its domain and range are all real numbers; and this is a continuous function.1792

OK, in Example 4, we are given f(x) = 3x2 - 4, and asked to find f(2), f(6), and f(2k).1805

First, f(2): recall that, when you are asked to find a function for a particular value of x,1824

you simply substitute that value for x in the equation; so f(2) equals 3(4) - 4, so that is 12 - 4; so f(2) = 8.1834

Next, I am asked to find f(6), and that is going to equal 3(62) - 4.1852

f(6) = 3(36) -4, and that turns out to be 108 - 4, so f(6) is 104.1860

Now, at first, this f(2k) might look kind of difficult; but you treat it just the same as you did with the numbers, when x is a numerical value.1877

Everywhere I see an x, I am going to insert 2k.1887

And figuring this out, 2 times 2, 2 squared, is 4; k times k is k2.1891

3 times 4 is 12, so I have 12k2 - 4; so f(2k) = 12k2 - 4.1902

So again, if you are asked to find the function of a particular value of x, you simply substitute whatever is given, including variables, for x.1910

That concludes this lesson of Educator.com; I will see you back here soon!1919