Determine the inverse of 𝑓 of 𝑥 equals a third 𝑥 plus two.
What’s it mean in general to find the inverse of a function like 𝑓? The inverse of 𝑓 is written like this: it’s an 𝑓 with a superscript minus one.
You might think that 𝑓 inverse of 𝑥 is just one over 𝑓 of 𝑥, but it’s not. Let’s see the proper definition. If 𝑦 equals 𝑓 of 𝑥, then 𝑥 equals 𝑓 inverse of 𝑦, and this defines 𝑓 inverse given 𝑓.
Notice that the second equation has variables in the opposite order to the way you might expect. The input to 𝑓 inverse is 𝑦 and the output from it is 𝑥.
The function 𝑓 takes an input 𝑥 and returns the output 𝑦. The function 𝑓 inverse takes the output from 𝑓, 𝑦, and returns its inputs to 𝑓, which generated that output 𝑥. The input to 𝑓, which we’ve called 𝑥, is the output from 𝑓 inverse. And the output from 𝑓, which we’ve called 𝑦, is the input to 𝑓 inverse.
Okay so let’s write down 𝑦 equals 𝑓 of 𝑥. Here we’ve used the definition of 𝑓 of 𝑥 to write 𝑦 equals a third 𝑥 plus two. And we’d like to rearrange this equation into the form 𝑥 equals something to do with 𝑦 because then we can just read off 𝑓 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 𝑥 is equal to three times 𝑦 minus two.
Now we can compare this with 𝑥 equals 𝑓 inverse of 𝑦 because 𝑦 is still equal to 𝑓 of 𝑥. So it’s clear what 𝑓 inverse of 𝑦 is. 𝑓 inverse of 𝑦 is three times 𝑦 minus two.
This determines the function 𝑓 inverse completely. So given an input 𝑦, we know that the output will be three times 𝑦 minus two, but of course it’s traditional to use 𝑥 as the input to a function, so we just change 𝑦 to 𝑥 and say that 𝑓 inverse of 𝑥 is three times 𝑥 minus two. And that’s our answer.