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Video: Finding the Inverse Function of a Given Linear Function

Alex Cutbill

Determine the inverse of 𝑓(𝑥) = (1/3) 𝑥 + 2.


Video Transcript

Determine the inverse of 𝑓 of π‘₯ equals a third π‘₯ 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 π‘₯ is just one over 𝑓 of π‘₯, but it’s not. Let’s see the proper definition. If 𝑦 equals 𝑓 of π‘₯, then π‘₯ 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 π‘₯.

The function 𝑓 takes an input π‘₯ 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 π‘₯. The input to 𝑓, which we’ve called π‘₯, 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 π‘₯. Here we’ve used the definition of 𝑓 of π‘₯ to write 𝑦 equals a third π‘₯ plus two. And we’d like to rearrange this equation into the form π‘₯ 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 π‘₯ is equal to three times 𝑦 minus two.

Now we can compare this with π‘₯ equals 𝑓 inverse of 𝑦 because 𝑦 is still equal to 𝑓 of π‘₯. 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 π‘₯ as the input to a function, so we just change 𝑦 to π‘₯ and say that 𝑓 inverse of π‘₯ is three times π‘₯ minus two. And that’s our answer.