### Video Transcript

Determine the inverse of π of π₯ equals a third π₯ plus two.

Whatβs it mean in general to find the inverse of a function like π? The inverse of π is written like this: itβs an π with a superscript minus one.

You might think that π inverse of π₯ is just one over π of π₯, but itβs not. Letβs see the proper definition. If π¦ equals π of π₯, then π₯ equals π inverse of π¦, and this defines π inverse given π.

Notice that the second equation has variables in the opposite order to the way you might expect. The input to π inverse is π¦ and the output from it is π₯.

The function π takes an input π₯ and returns the output π¦. The function π inverse takes the output from π, π¦, and returns its inputs to π, which generated that output π₯. The input to π, which weβve called π₯, is the output from π inverse. And the output from π, which weβve called π¦, is the input to π inverse.

Okay so letβs write down π¦ equals π of π₯. Here weβve used the definition of π of π₯ to write π¦ equals a third π₯ plus two. And weβd like to rearrange this equation into the form π₯ equals something to do with π¦ because then we can just read off π inverse.

So the first step is to subtract two from both sides. And now we can multiply by three, and finally we can swap both sides to get that π₯ is equal to three times π¦ minus two.

Now we can compare this with π₯ equals π inverse of π¦ because π¦ is still equal to π of π₯. So itβs clear what π inverse of π¦ is. π inverse of π¦ is three times π¦ minus two.

This determines the function π inverse completely. So given an input π¦, we know that the output will be three times π¦ minus two, but of course itβs traditional to use π₯ as the input to a function, so we just change π¦ to π₯ and say that π inverse of π₯ is three times π₯ minus two. And thatβs our answer.