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### Course: Precalculus (Eureka Math/EngageNY) > Unit 3

Lesson 12: Topic B: Lessons 16-17: Function composition- 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)
- Modeling with composite functions
- Modeling with composite functions: skydiving
- Model with composite functions

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# Modeling with composite functions

Sal determines the correct functions to compose (and the correct order) in order to model a given relationship, and vice versa.

## Want to join the conversation?

- So, could it be phrased as (H°T)(r)? Or how would it be formatted in that form?(7 votes)
- That is the correct format(5 votes)

- Why wouldn't the answer be T(H(k))?(1 vote)
- You need to match up the correct input/output values. Remember, the inner function is done 1st and then the output of the inner function is used as input to the outer function.

Sal is creating a function that defines the "height of the tree as a function of its altitude".

This means the input to the function is "altitude" and the output needs to be "height of the tree".

Which function has an output of "height of tree". Its H. So, this has to be the outer function.

Which function has an input of "altitude"? Its T. So, we have to use function T 1st. It creates as output "the average temperature", which happens to be the input for H.

If you tried to do T(H(k)), it won't work. Function H comes first. The input for H has to be temperature, but Sal needs an input that is "altitude".

Hope this helps.(16 votes)

- For the second question (4:48), wouldn't both expressions H(T(r)) and H(B(r)) represent the height of a tree as a function of its altitude?(3 votes)
- Not necessarily. The function B takes x (height of tree) as an input instead of r (altitude). Even if it did take r (altitude) as an input, B would output
**number of birds nesting**. However, that's not what H depends on: it depends on avg. temp., which is outputted by T instead.

That's a lot of "outputs" and "inputs" but I hope this helps!(2 votes)

- Why isn't the answer to the second question, H(k(T(r))) or H(T)?(2 votes)
- k is a variable, not a function, so the first option is nonsensical.

H(T) is actually correct, you just haven't written the input variable for T. There are contexts where it's okay to not write the input variable, and just write T instead of T(r), but it's best to maintain the habit of writing them out when you're still learning about function composition.(1 vote)

- For the second question about the trees, the first input is also the height of a tree, so why don't we use x? Is it because it is an input that was created with H(k), if so does H(k)=x(1 vote)
- The word problem said a function, H(k) is a function, not x(2 votes)

- How is H(T(r)) the answer? T(r) represents the "average temperature at that location" not altitude? Shouldn't the answer be H(r)?(1 vote)
- You are looking for height as a function of altitude, T(r) shows the average temperature at that location as a function of it's altitude, then you put that function into H, it is still a function of altitude though because T(r) is a function of altitude.(2 votes)

- How you solve f (x)= 2x-3 for rivers function(1 vote)
- This might be the wrong place to ask that question(2 votes)

- To clarify, when Mr. Khan says "the height of the tree as a function of its altitude" (5:19), what does it mean to be "as a function", and how does that correlate into having the input as the altitude?(1 vote)
- The phrase "the height of the tree as a function of its altitude" tells you the output of a function is the height and the function accepts altitude as its input. In other words, the functions will calculate the height of the tree when you input the tree's altitude.(2 votes)

- like there are inverse trig functions like arcsine(sin^-1) and cossine(cos^-1) is there something as a inverse function(1 vote)
- So there are inverse functions. An inverse function just inverse of a function. In fact sin^-1 is the inverse function of sine. You could also take the inverse of non-trigonometric functions.

There should be a section on Khan Academy.(2 votes)

- How would you interpret H(T(r))?(0 votes)
- You're inputting a location, r, and getting out a height of a tree there. So it would be the height of trees as a function of location.

We need to make sure that all of the intermediate steps make sense, but they do in this case. T accepts location and outputs temperature, and H accepts temperature and outputs height.(4 votes)

## Video transcript

- [Voiceover] "Carter has noticed a few "quantitative relationships related "to the success of his football team "and has modeled them with
the following functions." All right, this is interesting. So he has this function, which he denotes with the capital N and it's the winning, and the input of it is
the winning percentage, W and the output is the average
number of fans per game. So, he's making some
type of model that says, look the number of fans per game are gonna be in some way dependent on what your winning percentage is. And, I'm assuming he's modeled the higher the winning percentage, the more fans are gonna show up at a game. Now this is, W, the input is the average
daily practice time, x, and the output is the winning percentage. All right, that makes sense. Probably once again, probably some type of a positive
effect of practicing more is going to create a
higher winning percentage. And this other function, number of rainy days, r and then average practice time. Yup, well the more rainy days you have well that's going to lower
your average practice time. So, I definitely see how practice time, P, would be a function of
number of rainy days. "The expression N(W(x)) represents
which of the following?" Well before we even look at the choices, let's think about what's happening. This is another way of denoting we're gonna take x, we're gonna take x right over here, and we are going to input it into W and we're going to get out W(x) and then we're going to input
that into the function N. And, we are going to get out, N(W(x)). So, what does the function W do? What does the function
W do, right over here? Well, that's winning percentage as a function of practice time. So, you input practice, practice time, and it gives you, it somehow predicts a winning percentage, winning percentage. And, then you take that winning percentage and you input it into function N. Function N is going to output
the number of fans per game, based on winning percentage. So this is number of fans. So when you take the composite function, you're actually creating a function that starts with practice
time as the input and shows the number of fans that are gonna be dependent
on your practice time. So this is interesting. So, we should look for a choice that says, how does the number of
fans that show up at a game how is that dependent on practice time, x? All right. "The team's winning
percentage as a function "of the average daily practice time." Now that would be just W(x). If they said just W(x)
that'd be winning percentage as a function of average
daily practice time. So, I can cross that one out. The average number of fans per game, all right this is interesting because that's what the
final output's going to be in terms of the average
number of fans per game, that is the output of the function N, the function N right over here. "The average number of fans per game "as a function of the number
of rainy days in a season," Nope. We're not doing that. We're doing it as a
function of practice time. You could construct that. In fact, if you wanted to do this that would be N as a function of W as a function of P of r. So, that would have been this choice where you input the number of rainy days from that you're able to
figure out practice time and then you input practice time to figure out win perecentage and then you input win percentage to figure out the number
of fans in the crowd. But that's not what we're doing here. We're just starting
with daily practice time and getting to fans per game. So let me rule this one out. And if you found this one a little bit, what I just did a little bit confusing I encourage you to try to set up a diagram like I just did in the beginning. Instead of saying, oh well,
we could start with r to get, use that as input to get
average daily practice time and then use that as an input into W to get winning percentage. Then use that as an input into N to get average number of fans per game, but that's not what they're
describing for N(W(x)). "The average number of fans per game "as a function of the team's
average daily practice time." Yeah, that's what's going on. You have your average practice time, x being inputted into the function W. So your average practice
time is going inputted into W and it outputs winning percentage, which you then input into N to get the average
number of fans per game. The average number of fans per game as function of the team's
average daily practice time. So, yup, I definitely like that choice. Let's do another one of these. This is interesting. "Deniz studied the park near her home "where she identified several
quantitative relationships "and modeled them with
the following functions." So, B, it inputs the height
of a tree in terms of x and it outputs the number of
birds nesting in that tree. H, input the average temperature
at a specific location and it outputs the height of
the tree at that location. And T, the altitude of a specific location and then, if that's the input, and then the output is
the average temperature at that location. All right, this is interesting. "According to Deniz's findings, "which of the following
expressions represents "the height of a tree as a
function of its altitude?" So we want to figure out, we want to output the height of a tree and we want to input, the
altitude of a specific location. So, let's think about it. If we take our altitude
at a specific location, r and we input it into the function T, out of that we're going to get T(r). T, I'll be writing a little bit neater. We're gonna get T(r), which would represent average
temperature at that location, average temp, and then if we take the average
temperature at that location and input it into function H, and then we input it into function H, we are going to get the height
of a tree at that location. So, we're going to get H(T(r)) and so this is going to be
height of tree at that location, height of tree. And so, there you have it, H(T(r)). You start with r, altitude
at a specific location. Input it into function T. T's gonna spit out the average
temperature of that location. You input that into H. It's gonna get you the height
of the tree at that location. So, H(T(r)). H(T(r)) is this choice right over there.