# Thread: When A Function Is to the power of -1

1. ## When A Function Is to the power of -1

When A Function Is to the power of -1what does it mean?

2. Originally Posted by abc10
When A Function Is to the power of -1what does it mean?
It can have two meanings.

If you take the literal meaning for "power of -1", then

$\displaystyle f^{-1}(x) = \frac{1}{f(x)}$.

But the more-often used meaning for this notation is the "Inverse Function"

Basically, if you have a function $\displaystyle f(x)$, then the inverse function $\displaystyle f^{-1}(x)$ is the function whose graph is a reflection along the line $\displaystyle y = x$. So the $\displaystyle x$ and $\displaystyle y$ values have swapped. Also, the domain and range have also swapped.

3. Expanding slightly on what prove it said.

A function to the power one -1 is the inverse of that function.

the inverse of a number is a number that multiplied by the original equals 1.
$\displaystyle x^{-1}x=1$ always.

-----$\displaystyle \int$

4. ## Inverse of a function

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 $\displaystyle 2$.)