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## Precalculus

### Unit 1: Lesson 2

Modeling with composite functions# Meaningfully composing functions

Decide which composed functions make sense by checking that the value that one function passes to the other is the right kind of input. Created by Sal Khan.

## Want to join the conversation?

- What does the centimetre of precipitation mean?(0 votes)
- It means how many centimeters of rain is on the ground, for example in a flood it would be something like 100-200 centimeters of precipitation.(6 votes)

## Video transcript

- [Instructor] We're
told that Jaylen modeled the following relationships
about their bus ride. So there's three functions here. We have their inputs, and
we have their outputs. So function P, the input is
the time the bus arrives, given as lowercase b, and the output is the
probability that Jaylen gets to work on time. So P of b, all right. Function N, the input is
the time the bus arrives, given as k, and the output is the number
of people at the bus stop when the bus arrives. So it's the number of
people at the bus stop as a function of the time the bus arrives. And then T, centimeters of precipitation, the input is centimeters of
precipitation per hour, x. And this gives us the time the bus arrives as a function of that precipitation. Interesting. Now, they ask us which of the
following composed functions makes sense in this context? And we have to pick two. So pause this video and see
if you can have a go at it before we work through it together. All right, now let's look at this first composite function here. So let's see what's going on. One way to think about
this is we're taking x, we are inputting that
into our function, T, that will then output T of x, and then we're trying to input that into our function P, which would then output P of the input, which is T of x. Now, does this make sense? X is the centimeters of
precipitation per hour. T of x is a time the bus
arrives as a function of that. So we're gonna take the
time the bus arrives, is that a reasonable input into P? Well, let's see, function P, the input is the time the bus arrives. So this makes a lot of sense. We're taking this, which
is a time the bus arrives, and we're using that as an input into P, which is exactly what we
want to be the input into P. And so we have P as a function
of the time the bus arrives, which is then a function of the centimeters of
precipitation per hour. So I like this choice right over there. Now let's look at the next choice. P as a function of N of k. Let's see, so we're taking k, which is the time the bus arrives, we're inputting it into the function N, which then outputs N of k. Now the output here, N of k, is the number of people at the bus stop when the bus arrives. And then we're trying to put
that into our function P. Now, does that make sense
to take the number of people at the bus stop when the bus arrives and input it into a function
that expects as the input the time the bus arrives? That does not make sense. P wants as an input time the bus arrives, but we're giving it the number
of people at the bus stop. So I do not like this choice. Now let's look at choice C. So T of x, right over here, we already know that outputs
the time the bus arrives, which we know is a legitimate
input into function N. Function N takes the time the bus arrives and as a function, and
based on that input, it gives you the number
of people at the bus stop. So this makes sense. So I like this choice as well. So I'm guessing I won't like choice D, but let me validate that. So P of b. So this is going to
give us the probability that Jaylen gets to work on time, and then we're inputting
that into this function, T, that's expecting centimeters
of precipitation per hour. Well, that's not going to work out. You're trying to take a probability and you're inputting it into a function that expects centimeters
of precipitation per hour, so I would rule that one out as well. So I like A and C.