WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take a look at an introduction to functions.
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There is quite a bit to cover when it comes to functions, we will go over this bit by bit.
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Some of the first things I will do with functions is look at some of the terms.
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In other words, what exactly makes a function and why is it a special type of a relationship.
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What is it that makes a function a function?
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We will get into a term such as the independent and dependent variable so they can find those parts in a function.
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We will talk about how functions can be like little machines and how they have input and output.
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We will call this out its domain and its range.
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We will get into a little bit on how you can represent function using diagrams that we can better keep track of the inputs and outputs
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and also how you can work with functions by evaluating them.
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Let us take a look.
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A function is a very special type of relation.
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What a relation does is it connects 2 variables and usually we use a pair of ordered pairs to describe that connection.
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Maybe I have an x variable and a y variable and one other connected.
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I can show you that anytime I use a 2 for x, I get a 3 for y.
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There are many other types of connections that you can consider as part of a relation.
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Some of our equations can be considered a type of relation.
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A function is just a special kind of one of these relations.
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It is special because for every x value it has, there is one and only one y value associated with it.
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What that means is that if you use a particular x value, you are never going to get two different values using that one x value.
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When we represent functions, we usually identify some sort of independent variable.
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This is what we get to choose for in terms of the x.
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The dependent variable is usually represented by y and it depends on whatever we choose for x.
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Some of the notation you may see for functions looks like this.
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It is important that we can pick out each of the different parts here.
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The first thing is what is this f out in front? f represents the name of the function.
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This function is called f.
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The x inside the parentheses right next to it is not multiplication, what that is indicating is what the variable is.
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We have our function called f and it takes a variable of x.
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Everything after that on the other side, that represents the relation or how things are connected.
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You can also consider this as the rule of the function or what it does when you use a particular x.
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The way you read this notation is f(x) = 3x + 9.
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Our f part helps us identify what variables are being used inside the function.
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To better understand what type of relations our function is, I like to tell my students that they are a lot like little machines.
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They take some input and you put into and they produce some output.
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A nice little diagram would look something like this down here.
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Maybe my function f here takes an input of 3 and produces an output of 5.
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If we start to gather up together more of its inputs and outputs, we can represent those as a whole bunch of ordered pairs.
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What this list of ordered pairs would say is that my function takes an input of 1 and produces an output of 2.
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If I give it an input of 3 then I get an output of 5.
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Every ordered pair is telling me what the input and output was for that particular value.
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Now, if we collected together all of the inputs, this would be known as the domain of the function.
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If we collected together all of the outputs of the function, this be known as the range.
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Using the example from a whole list of ordered pairs, we can actually identify all the things used as an input and all the things used as an output.
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Remember this first value here, those are our x’s, those are inputs and the second values, those are our outputs.
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Let us simply just list out all of the inputs and call that our domain and list out all the outputs and call that our range.
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We used 1, 3, 4, 7 for input and what we got as output was 3, -2, 5, 10.
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You can also represent a relation in the same way, so it is important to point out that this particular relation here is actually a function.
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The reason why we know it is a function is that every time we are looking at any particular output, it only has one output associated with it.
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Input of 1 output of 3, input of 3 output of -2.
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You do not see anything on this list like 1, 7 because that would be a problem
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because the 1 would go to 3 and 1 would go to 7, you will get 2 outputs.
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I just want to make a little note here that this is a function because it satisfies our definition.
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A great way to visualize what is happening with the inputs and outputs is to use a diagram.
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One way to do that is to list all of your domains and group it together in a circle or a box.
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And you group together all of your range in the same way.
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You can show the relation by using arrows to connect the two.
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That is exactly what I have done down below.
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Here I have two different relations, you can see my inputs and you can see my outputs.
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We will consider these our domain and our range.
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The big question is does it satisfy the definition of a function or is it just simply some relation.
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Let us take a look at it and see what happens to those inputs and outputs.
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Remember, it is a function if every input you give it goes to exactly 1 output.
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Let us see what we got.
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Looking at this input here of 1, it looks like it has an output of 4 and that is the only thing it has for now, I will put just 4.
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Let us check out the 3, if I use an input of 3 it looks like I will get an output of 4 and I will get an output of 0.
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Let me highlight that.
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Notice how this one has an arrow that goes right over here to 4 and has another one that goes out to 0.
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When I use an input of 3, it goes to two different outputs then I can say that this is not a function.
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It is close but it does not work.
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Since we know that one was not a function, let us take a look at the other relation.
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When I use an input of 1 it goes to 8, I will use an input of 3 it goes to 5, input of 6 goes to 1, input of 7 goes to 0.
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Every time I use an input here, it only goes to one output.
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This one was okay, this one is a function.
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You can see these diagrams help out with looking at what is in the domain and range, it is all grouped together nice and simple.
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In some functions, they may not actually specify what domain is being used and do not actually says what the domain is.
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We just assume that is known as the natural domain.
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What this natural domain represents is all of the possible real numbers that you could use as input
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as long as it does not make the function itself undefined.
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You might start with thinking about all possible numbers and eliminating ones that you can not use.
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To help better identify what you can not use in the domain, look for stuff like fractions or even roots.
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The reason for this is, you cannot divide by 0.
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If any value would make the bottom of a fraction 0 then you toss them out, they are not in the domain.
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Also, we do not want to have to deal with imaginary numbers, we do not want negative numbers underneath an even root.
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If that happens, we will toss those out of the domain as well.
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When it comes to writing down a lot of different numbers in the domain, there is a special way that we go about doing that.
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We often use what is known as interval notation.
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What interval notation is, it is a way of packaging up all of the range of numbers on a number line.
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Let me give you a nice quick example.
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I wanted to describe all the numbers between 2 and 3.
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I could do that by shading in those numbers on a number line so you can see that I have shaded in 2, 3, and everything in between.
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To use interval notation to do this, I would list out the starting point and the endpoint of everything that I have shaded.
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Then I would use a bracket to say that the endpoints are included.
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Everything between 2 and 3, I’m using that square bracket because 2 and 3 shall also be included.
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If we can go all the way up to a point but maybe not included, we will still use interval notation for this.
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But usually we use a parenthesis when it is not included.
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I have re adjusted the number line here, now I shaded everything from to 2 to 3 but not including 3.
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I have made the corresponding interval notation to reflect that.
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I started at 2 stop at 3, includes 2, but does not include 3.
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How we work with these functions?
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One of the more common things we do with functions is to simply evaluate them.
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To evaluate a function, what we simply do is substitute in the value given in place of x.
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If I'm dealing with this function f(x) = 3x + 9 and maybe I want to evaluate it at the number 7.
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The way I do this is I just replace all x’s with 7.
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You will see that the process itself is not that difficult.
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It is sometimes just the notation that throws people off because when you look at the left side over here,
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your brain sometimes looks at that and you want to think of multiplication, but it is not multiplication.
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When you have that f(7), what you are saying is that you used an input of 7 and then over here is what you got as your output.
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Keep that in mind when you are evaluating functions.
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Let us get into some examples and see if we can figure out whether certain relations are functions
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and maybe identify a domain, range, and all that fun stuff.
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In this relation, I can see that my inputs would be the first values and my outputs those are the second values.
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Let us represent this using a diagram just to help out.
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We will group together all of our inputs into a giant circle over here.
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We will group together all of our outputs into a giant circle over here.
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First the input, I have -3, 6, I have another -3 so I do not want to list it twice and I have 5.
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Our outputs are 7, -2, 4, and 9.
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When I use an input of -3 it goes over here to 7 and when I use an input of 6, this heads over to -2.
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I have another -3, -3 also goes to 4 and 5 goes to 9.
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That is a bit of a problem isn’t it?
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Notice how we have a single input and it goes to two different spots.
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If we have a single input going to 2 outputs then we can say that this is not a function.
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Here I see another diagram.
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Let us see if we can figure out if this one is a function, 13 goes to 7, 81 goes to 7, 32 goes to 60, 27 goes to 19 and 45 goes to 55.
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This one actually looks pretty good.
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This is a function.
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There is an interesting feature here that I want to point out,
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you noticed in the last one we had one input that went to 2 different outputs and that made it not a function.
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You may be curious if this would also make it not a function? After all, 13 and 81 go to the same spot.
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That is okay, you can have two inputs go to the same spot
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but what you do not want to happen is to have one input split off into two different places.
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This right here, this is okay.
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Let us see if we can identify the domain of a function just from looking at its equation.
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Since no domain is specified, we are going to look for the natural domain.
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That means we will assume that all real numbers can be used unless it makes the function undefined.
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In these examples, I put one with a fraction because we want be concerned about dividing by 0.
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I put another one with a square root because we do not want negatives underneath an even root, like the square root.
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In this first one, the way that it would not work out is if I had a 0 in the bottom.
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Are there any restricted values for x?
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Anything can x can not be would that give us a 0.
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Well, if I had to solve just the bottom part, I can see that if x was equal to -1 I would definitely have a problem.
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I think that is the only issue that we would end up with.
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Let us just write them out, x can not be -1.
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Anything else is perfectly acceptable.
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If I was looking at a number line and trying to shade in all the different things x could or could not be,
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the number line would look something like this.
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It could definitely be any of these negative values over here.
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I’m digging a big open hole at -1 since it can not be that and it could be anything greater than that.
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It can be any value but not just -1.
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I will represent that using some intervals.
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It could be anything from negative infinity all the way up to -1, I would not include that.
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It can be anything from -1 up to infinity, I will use little u’s to connect those.
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It can be anything just not -1.
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Let us try another one, f(x) = √x+ + 5.
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I want to make sure that whatever x + 5 is, we do not want it to be negative.
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It is okay if it is 0 but definitely we do not want it to be negative.
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The only way it would end up being negative is if x was less than -5.
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Let us say x can not be less than -5.
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We will represent this on a number line as well so we can end up making an interval for it.
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Let us see, here I have -5, -4, -3, -2, -1, 0, I have some things that are less than -5.
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x can be anything, it can even be -5 because you can take the square root of 0, it can be anything greater.
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The domain of this one, the natural domain would be from -5 up to infinity and it is okay to include that -5.
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For this one, we want to work on evaluating the function for some given values of x.
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Be careful on this notation over here, we are not multiplying or simply substituting 3 into our function and seeing what the result is.
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I'm going to replace everywhere I see an x with this number 3.
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Once I substituted it in there, we go through and clean it up a bit, I have -12 + 9 or -3.
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Our input was 3 and our output is -3.
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Let us do the same thing for f(-5).
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Same function, we will just put in a different value, -5.
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We will work to simplify it.
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-4 × -5, negative × negative will be positive, 4 × 5 is 20, 20 + 9 and we would get 29.
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In evaluating functions, just simply substitute a value in there for x and end up simplifying it.
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That is all I have for now, thank you for watching www.educator.com.