Hello abc10
Just to expand a little further on the replies so far:
Yes, $\displaystyle f^{-1}$ denotes the inverse of the function $\displaystyle f$.
Let's take a simple function like $\displaystyle f(x) = 2x + 5$. Then, for example:
$\displaystyle f(0) = 5$
$\displaystyle f(1)=7$
$\displaystyle f(2)=9$
... and so on.
Then $\displaystyle f^{-1}$ is the function that reverses this process. So:
$\displaystyle f^{-1}(5) = 0$
$\displaystyle f^{-1}(7)=1$
$\displaystyle f^{-1}(9) = 2$
... and so on.
Perhaps you can work out what $\displaystyle f^{-1}$ does? Since $\displaystyle f$ multiplies by $\displaystyle 2$ and then adds $\displaystyle 5$, $\displaystyle f^{-1}$ will do the 'opposite' things in the reverse order. In other words, $\displaystyle f^{-1}$ will:
- subtract $\displaystyle 5$; and then
- divide by $\displaystyle 2$.
So:
$\displaystyle f^{-1}(x)= \frac{x-5}{2}$
(Try this function with the values $\displaystyle 5$, $\displaystyle 7$ and $\displaystyle 9$ and check that you get $\displaystyle 0$, $\displaystyle 1$ and $\displaystyle 2$.)
Grandad