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## Precalculus

### Course: 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

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

## Want to join the conversation?

- I'm confused when it comes to how to solve questions involving functions that look something like: "If f(x) = -2x + 5, what is f(-3x) equal to?". For reference this is a problem on the 4th SAT practice test here on Khan Academy that I got
**right**, only I don't understand the real logic behind it. If anyone could give me some intuition I would be thankful manyfold :)(14 votes)- Hallo

Well first of all we can look at this problem and say, well what do we see in common with f(x) and f(-3x)? All you are doing is substituting the x and the -3x. You have to do the same thing to both sides of the equation so just put a -3x for every x in -2x + 5. Once you do this you will get: -2x(-3x)+5. Then it's just simple multiplication to get 6x^2 + 5. So, f(-3x) = 6x^2 + 5

Hope that helped,

Silas(16 votes)

- why/when would you use this(4 votes)
- Computer programming breaks algorithms down into reusable functions / modules.

They would use this concept of composite functions where the output of one function becomes the input to another function.

A real life example would be in calculating what you pay at a store.

Let's say you buy a sweater and it is on sale for 25% off. And, you live in a state where they apply income tax to your purchase. You could have 2 functions:

1) Calculate the discounted price: Input = original price; output = discounted price

2) Apply tax: Input = discounted price; output = price with tax(22 votes)

- At1:07what does it mean if h is "of" g? I have never heard the "of" term used before.(5 votes)
- Ethan,

h "of" g simply means the same as it would when you said h "of" x or h "of" 2.

I hope that answers your question.(12 votes)

- I am really confused and do not know where to start Please Help!!

~Naomi(8 votes) - can someone help me? I just watched this video and when i went to the activity i lost all thought. Anyone have tips on how to stop that?(3 votes)
- Just calm down and rethink that is what I did at first but the more you see and practice the more better you will be(3 votes)

- What would be the next step for g(f(0.25)?(2 votes)
- You haven't given the definition of the functions, so there is no way to give you specifics.

Your basic approach would be to :

1) Find f(0.25)

2) Then use the result as your input to g(x)(10 votes)

- Quick and slightly unrelated question: What program does Sal use for these types of videos? As in, the one he uses to draw for us to see what he means.(3 votes)
- Sal Khan, the founder of Khan Academy, uses a digital tablet and stylus to create the videos in which he writes and draws. He uses a software called SmoothDraw to write and draw on the digital tablet.

SmoothDraw is a freehand drawing and painting software that is specifically designed for use with digital tablets. It provides a variety of drawing tools and options, including pens, brushes, shapes, and text. It also includes features such as layer support, image resizing, and color management.

Using SmoothDraw, Sal Khan is able to write and draw on his digital tablet in real-time, which allows him to explain concepts and work through problems visually in his videos. This has been a key factor in the success of Khan Academy, as it allows students to see and understand concepts in a visual and interactive way.(6 votes)

- Why did he continue to solve the problem after he got the answer of 3?(1 vote)
- He continued to solve the problem after he got the answer of 3 because in a composite function problem, there are two steps. The first step is to find the answer of the function within the function. The second step is to input the answer of the function of the function in to the function on the outside. This is the best I can explain it, and I hope it helped!(10 votes)

- im confused about questions where it says something like f(g(x)) and then says the function equation thing. Could somebody explain this to me.

Thank You(4 votes)- That is the expression of the function first you find g(x) it will give you answer and let’s that answer is y then you do f(y) then you will get your final outcome or your final answer. I hope this answers your question(3 votes)

- is there an order in which one you have to substitute first(3 votes)
- You'd want to solve for the one which you have a solution for, then you would be able to figure out the rest of the equation(4 votes)

## 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.