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## Precalculus

### Unit 1: Lesson 1

Composing functions- Intro to composing functions
- Intro to composing functions
- Composing functions
- Evaluating composite functions
- Evaluate composite functions
- Evaluating composite functions: using tables
- Evaluating composite functions: using graphs
- Evaluate composite functions: graphs & tables
- Finding composite functions
- Find composite functions
- Evaluating composite functions (advanced)

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# Evaluating composite functions

CCSS.Math:

Sal evaluates (h⚬g)(-6) for g(x)=x²+5x-3 and h(y)=3(y-1)²-5.

## Video transcript

- [Voiceover] So, we're told that g of x is equal to x squared
plus 5 x minus 3 and h of y is equal to 3 times y minus 1 squared, minus 5. And then, we're asked, what is h of g of negative 6? And the way it's written might
look a little strange to you. This little circle that we have
in between the h and the g, that's our function composition symbol. So, function, function composition, composition, composition symbol. And one way to rewrite this, it might make a little bit more sense. So, this h of g of negative 6. You could rewrite this as, this is going to be the same thing as g of negative 6, and then h of that. So, h of g of negative 6. Notice, I spoke this out the
same way that I said this. This is h of g of negative 6. This is h of g of negative 6. I find the second notation
far more intuitive, but it's good to become familiar with this function composition
notation, this little circle, because you might see that sometime and you shouldn't stress, it's
just the same thing as what we have right over here. Now, what is h of g of negative 6? Well, we just have to remind
ourselves that this means that we're going to take
the number negative 6, we're going to input
it into our function g, and then that will
output g of negative 6, whatever that number is, we'll
figure it out in a second, and then we're going to input
that into our function h. We're going to input
that into our function h. And then, what we output is going to be h of g of negative six, which
is what we want to figure out. h of g of negative 6. So, we just have to do
it one step at a time. A lot of times, when you
first start looking at these function composition,
it seems really convoluted and confusing, but you just have to, I want you to take a breath
and take it one step at a time. Well, let's figure out
what g of negative 6 is. It's going to evaluate
to a number in this case. And then, we input that
number into the function h, and then we'll figure out another -- that's going to map to another number. So, g of negative 6. Let's figure that out. g of negative 6 is equal to negative 6 squared, plus 5 times negative 6, minus 3, which is equal to positive
36, minus 30, minus 3. So, that's equal to what? 36 minus 33, which is equal to 3. So, g of negative 6 is equal to 3. g of negative 6 is equal to 3. g of negative 6 is equal to 3. You input negative 6
into g, it outputs 3. And so, h of g of negative
6 has now simplified to just h of 3 because
g of negative 6 is 3. So, let's figure out what h of 3 is. h of 3 --. notice,
whatever we outputted from g, we're inputting that now into h. So, that's the number 3,
so h of 3 is going to be 3 times 3 minus 1, 3 minus 1 squared, minus 5, which is equal to 3 times 2 squared, this is 2 right over here, minus 5, which is equal to 3
times 4 minus 5, which is equal to 12 minus 5, which is equal to 7. And we're done. So, you input negative
6 into g, you get 3. And then, you take that output from g and you put it into h and you get 7. So, this right over here is 7. All of this has come out
to be equal to 7. So, h of g of negative
6 is equal to 7. h of g of negative 6 is equal to 7. Input negative 6 into
g, take that output and input it into h, and
you're gonna get 7.