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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.

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• So, could it be phrased as (H°T)(r)? Or how would it be formatted in that form?
• That is the correct format
• 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.
• For the second question (), wouldn't both expressions H(T(r)) and H(B(r)) represent the height of a tree as a function of its altitude?
• 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!
• Why isn't the answer to the second question, H(k(T(r))) or H(T)?
• 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
• 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.
• How you solve f (x)= 2x-3 for rivers function
(1 vote)
• This might be the wrong place to ask that question
• To clarify, when Mr. Khan says "the height of the tree as a function of its altitude" (), 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.
• 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.