### Video Transcript

Given π of π₯ equals three π₯ minus one and π of π₯ equals π₯ squared plus one, which of the following expressions gives π of π of π₯? Is it option A) three π₯ squared plus three, option B) nine π₯ squared minus six π₯ plus three, option C) three π₯ squared plus two, option D) three π₯ squared, or option E) nine π₯ squared minus six π₯ plus two.

So what is the thing here which is represented by this π and then this small circle and then this π? Well, itβs a new function which you get by composing the functions π and π. We can explain what this function does by showing its effect on an arbitrary input π₯. We have that π of π₯ is equal to three π₯ minus one and π of π₯ is equal to π₯ squared plus one and weβd like to have a similar expression on the right-hand side to show what happens to π₯ when π of π is applied instead of π or π alone.

How do we do this? Well, we use a definition of function composition which tells us that π composed with π, or π of π, of π₯ is equal to π of π of π₯. Before we try to find an expression for π of π of π₯, letβs just see where π of π takes the number two. And of course, we just replace π₯ by two wherever we see it in the above definition. So we get that π of π of two is equal to π of π of two. We want to evaluate the right-hand side, and we do that by first finding π of two and substituting that it.

What is π of two? Well, we can find that by using our expression for π of π₯. π of π₯ is π₯ squared plus one, so π of two is two squared plus one which is five. And we substitute that in. So now we see that π of π of two is π of five. Now I need to find π of five. How do we do that? We use the expression that we have for π of π₯. π of π₯ is three π₯ minus one and so π of five is three times five minus one which is 14. So when the function π of π is given the input two, we get the output 14.

How do you find π of π of π₯ though? Itβs going to be very similar to finding π of π of two except weβre going to be dealing with algebraic expressions instead of numbers. So first again, we evaluated the right-hand side, and we do that by first finding π of π₯. Well, we have an expression for π of π₯ in the question, so letβs just substitute that expression in. So we get π of π₯ squared plus one.

Okay. Now what do we do with that? We use the definition for π, so π of π₯ is equal to three π₯ minus one. What is π of π₯ squared plus one then? Well, π takes an input and returns three times that input minus one. So the answer is: three times π₯ squared plus one minus one. And we can simplify this by first applying the distributive property to multiply out the brackets and then collecting the constant terms to get three π₯ squared plus two.

Okay. So now that we have our answer, letβs go through the options, and we can see that C, three π₯ squared plus two, is our correct answer.