Notation for inverse of a function

Question

Say we have $f(x) = y$. How are we going to write the inverse for this? Is it $f^{-1}(x)$ or $f^{-1}(y)$? Which is the correct one? If $f(x) = 3x+2$ what does $f^{-1}(x)$ mean?

Answer 1

You can write it as $f^{-1}(x)=\frac{x-2}{3}$, or as $f^{-1}(y)=\frac{y-2}{3}$, or as $f^{-1}(t)=\frac{t-1}{3}$. The variable doesn't matter, so long as you're consistent.

Answer 2

I think it is a little neater to write $f(x) = y \implies f^{-1}(y) = x$

$y = f(x) = 3x +2\\ x = f^{-1}(y) = \frac {y-2}{3}$

However once you have the function, the variable names become arbitrary.

$f^{-1}(y) = \frac {y-2}{3}\implies f^{-1}(a) = \frac {a-2}{3}, f^{-1}(q) = \frac {q-2}{3}, f^{-1}(x) = \frac {x-2}{3}$, etc.

Answer 3

If $f(x)=y$ then you may conclude (Assuming that the inverse exists) that $f^{-1}(y)=x$.

How do you find an inverse for a function like $f(x)=3x+2$?

You split the function to it's components, what $f$ does? It does two operations:

1. It multiplies $x$ by $3$.
2. It adds $2$.

Then you find the inverse of each operation but in the opposite order, so what's the inverse of addition is subtraction, hence we need to subtract $2$. And what's the inverse of multiplication is division hence we need to divide by $3$. We conclude that the inverse function subtracts $2$ and then divides by $3$ in other words $$f^{-1}(x) = (x-2)/3$$