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Given "g" of "x" is equal to eight "x" plus two

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find "g" of zero, "g" of negative five and "g" of

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two "x" plus three

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So all this is when they say "g" of "x" is equal

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to eight "x" plus two. This is a function definition

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They're saying you give me any "x"

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and you input it into my function of "g"

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and my function "g" will output

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It will associate with with that number "x"

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another number that happens to

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be eight "x" plus two

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This is one way to create an association,

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it's going to associate. If you were to put,

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If you were to put one here, if this "x"

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is one, you're going to do eight times one

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plus two. So you would associate one

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through "g", you would assocate one with

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eight times one plus two which is ten.

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So, its just another way of creating an association

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that you could do with sets of ordered pairs

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or you could do it graphically. But this is one of

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the most typical ways you that you will see a

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function defined. It says look, input an "x"

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over here and I'm going to output eight "x" plus two

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So with that out of the way lets think about what

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the "g" of zero is. "G" of zero literally just means

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we're inputting zero into this function and this

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function says, whatever your inputting, I'm going

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output eight times that plus two. So "g" of zero

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is going to be, let me write it this way, so it

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becomes clear what I'm doing

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"G" of zero is going to be eight times zero

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eight times zero, eight times zero plus two, plus

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plus two. So where ever we saw the "x"

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we replaced it with whatever we're inputting

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that is our "x" in this situation right over here

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and so this is equal to eight times zero plus two

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is just equal to two. So

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"g" of zero is equal to two, let me just write that

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right here "g" of zero is equal to two

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Now lets try "g" of negative five

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so let me write it this way "g" of... I'll write "g" of negative

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five is equal to, I'll try to do the color coding

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a little bit better this time. It's going to be equal

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to eight times our "x" which in this case is negative

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five, eight times negative five plus two, eight times negative five plus two

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we're inputting a negative five and we're going

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to output eight times negative five plus two

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wherever we saw the "x" we just replaced

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it with a negative five. And this is just equal

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to eight times negative five

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is negative forty plus two is equal to negative

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thirty-eight. So I'll write this over here on the right

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We have "g" of negative five is equal to

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negative thirty-eight. Now this

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last one is a little bit interesting,

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So we just did this one,

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now let's do "g" of... "g" of two "x" plus three.

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So now they're not inputting a number, they're

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inputting an expression

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and what we have to do, everywhere where we see

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a "x" here, we're going to replace it with that

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expression. If I did "g" of a circle

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Let me write this down, if I did "g" of

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an orange circle, an orange circle right over here

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By this function definition, by this function definition

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if I'm inputting an orange circle, I should output

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an orange circle eight times

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the orange circle, eight times the orange circle

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which doesn't make a lot of sense

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but hopefully you get the idea that whatever you're

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inputting you're just going to multiply it times eight and add two

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so if we input two "x" plus three we're going to output

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this times eight plus two

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wherever we see an "x" we replace it

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with this right over here. So let's try that out

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so now we have "g" of

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two "x" plus three, "g" of two "x" plus three

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is going to be equal to...

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is going to be equal to eight times whatever

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we're inputting, eight times whatever were inputting

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plus two. If we're inputting an orange, if this was

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orange, we'd put an orange here, if this is a negative

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five here we'd put a negative five here, if this is a zero

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we'd put a zero here. In this case, we're inputting two "x"

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plus three into our function, so we put two "x" plus three

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where the "x" is right over there. So two "x" plus

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three. Whatever we input this function says

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multiply it by eight and add two

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we're inputting two "x" plus three, multiply it by

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eight and then add two, and so this is going to be equal

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to, we just distribute the eight... we just distribute the

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eight and you get sixteen "x" plus

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twenty-four and you still have this plus two out over here

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you still have this plus two out over here

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so this is going to be equal to sixteen "x", sixteen

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"x" plus these two, you add them up you get

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twenty-six. So we can write over here and I'll do

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that same color so to be consistent

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We have "g" of two "x" plus three is equal to

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sixteeen "x" plus twenty-six and we're done.

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I know this can be confusing because you're

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inputting, you're replacing "x"

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with something that has "x" in it.

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But you really should just view this as if

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this were an orange ball or a star

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or even if this had "y's" in it

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Wherever you see an "x" you replace it

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with whatever you are inputting it. So in every

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instance we replace the "x" with two "x" plus three.