Meaningfully composing functions | Composite and inverse functions | Precalculus | Khan Academy

We're told that Jalen 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 Jalen gets to work on time, so P of b.

Alright, 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.

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. So 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.

Alright, 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 the time the bus arrives as a function of that. So we're going to take the time the bus arrives. Is that a reasonable input into P?

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 the 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.

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 the 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, 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: this is going to give us the probability that Jalen 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.