When A Function Is to the power of -1

• Dec 28th 2009, 07:07 PM
abc10
When A Function Is to the power of -1
When A Function Is to the power of -1what does it mean?
• Dec 28th 2009, 07:28 PM
Prove It
Quote:

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

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

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

Basically, if you have a function $f(x)$, then the inverse function $f^{-1}(x)$ is the function whose graph is a reflection along the line $y = x$. So the $x$ and $y$ values have swapped. Also, the domain and range have also swapped.
• Dec 28th 2009, 09:54 PM
integral
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.
$x^{-1}x=1$ always.

----- $\int$
• Dec 29th 2009, 01:39 AM
Inverse of a function
Hello abc10

Just to expand a little further on the replies so far:

Yes, $f^{-1}$ denotes the inverse of the function $f$.

Let's take a simple function like $f(x) = 2x + 5$. Then, for example:
$f(0) = 5$

$f(1)=7$

$f(2)=9$

... and so on.
Then $f^{-1}$ is the function that reverses this process. So:
$f^{-1}(5) = 0$

$f^{-1}(7)=1$

$f^{-1}(9) = 2$

... and so on.
Perhaps you can work out what $f^{-1}$ does? Since $f$ multiplies by $2$ and then adds $5$, $f^{-1}$ will do the 'opposite' things in the reverse order. In other words, $f^{-1}$ will:

• subtract $5$; and then

• divide by $2$.

So:
$f^{-1}(x)= \frac{x-5}{2}$
(Try this function with the values $5$, $7$ and $9$ and check that you get $0$, $1$ and $2$.)