Ex 1: Evaluating a function Evaluating Functions
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- Given "g" of "x" is equal to eight "x" plus two
- find "g" of zero, "g" of negative five and "g" of
- two "x" plus three
- So all this is when they say "g" of "x" is equal
- to eight "x" plus two. This is a function definition
- They're saying you give me any "x"
- and you input it into my function of "g"
- and my function "g" will output
- It will associate with with that number "x"
- another number that happens to
- be eight "x" plus two
- This is one way to create an association,
- it's going to associate. If you were to put,
- If you were to put one here, if this "x"
- is one, you're going to do eight times one
- plus two. So you would associate one
- through "g", you would assocate one with
- eight times one plus two which is ten.
- So, its just another way of creating an association
- that you could do with sets of ordered pairs
- or you could do it graphically. But this is one of
- the most typical ways you that you will see a
- function defined. It says look, input an "x"
- over here and I'm going to output eight "x" plus two
- So with that out of the way lets think about what
- the "g" of zero is. "G" of zero literally just means
- we're inputting zero into this function and this
- function says, whatever your inputting, I'm going
- output eight times that plus two. So "g" of zero
- is going to be, let me write it this way, so it
- becomes clear what I'm doing
- "G" of zero is going to be eight times zero
- eight times zero, eight times zero plus two, plus
- plus two. So where ever we saw the "x"
- we replaced it with whatever we're inputting
- that is our "x" in this situation right over here
- and so this is equal to eight times zero plus two
- is just equal to two. So
- "g" of zero is equal to two, let me just write that
- right here "g" of zero is equal to two
- Now lets try "g" of negative five
- so let me write it this way "g" of... I'll write "g" of negative
- five is equal to, I'll try to do the color coding
- a little bit better this time. It's going to be equal
- to eight times our "x" which in this case is negative
- five, eight times negative five plus two, eight times negative five plus two
- we're inputting a negative five and we're going
- to output eight times negative five plus two
- wherever we saw the "x" we just replaced
- it with a negative five. And this is just equal
- to eight times negative five
- is negative forty plus two is equal to negative
- thirty-eight. So I'll write this over here on the right
- We have "g" of negative five is equal to
- negative thirty-eight. Now this
- last one is a little bit interesting,
- So we just did this one,
- now let's do "g" of... "g" of two "x" plus three.
- So now they're not inputting a number, they're
- inputting an expression
- and what we have to do, everywhere where we see
- a "x" here, we're going to replace it with that
- expression. If I did "g" of a circle
- Let me write this down, if I did "g" of
- an orange circle, an orange circle right over here
- By this function definition, by this function definition
- if I'm inputting an orange circle, I should output
- an orange circle eight times
- the orange circle, eight times the orange circle
- which doesn't make a lot of sense
- but hopefully you get the idea that whatever you're
- inputting you're just going to multiply it times eight and add two
- so if we input two "x" plus three we're going to output
- this times eight plus two
- wherever we see an "x" we replace it
- with this right over here. So let's try that out
- so now we have "g" of
- two "x" plus three, "g" of two "x" plus three
- is going to be equal to...
- is going to be equal to eight times whatever
- we're inputting, eight times whatever were inputting
- plus two. If we're inputting an orange, if this was
- orange, we'd put an orange here, if this is a negative
- five here we'd put a negative five here, if this is a zero
- we'd put a zero here. In this case, we're inputting two "x"
- plus three into our function, so we put two "x" plus three
- where the "x" is right over there. So two "x" plus
- three. Whatever we input this function says
- multiply it by eight and add two
- we're inputting two "x" plus three, multiply it by
- eight and then add two, and so this is going to be equal
- to, we just distribute the eight... we just distribute the
- eight and you get sixteen "x" plus
- twenty-four and you still have this plus two out over here
- you still have this plus two out over here
- so this is going to be equal to sixteen "x", sixteen
- "x" plus these two, you add them up you get
- twenty-six. So we can write over here and I'll do
- that same color so to be consistent
- We have "g" of two "x" plus three is equal to
- sixteeen "x" plus twenty-six and we're done.
- I know this can be confusing because you're
- inputting, you're replacing "x"
- with something that has "x" in it.
- But you really should just view this as if
- this were an orange ball or a star
- or even if this had "y's" in it
- Wherever you see an "x" you replace it
- with whatever you are inputting it. So in every
- instance we replace the "x" with two "x" plus three.
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