# Inverse With Restricted Domain

• Nov 18th 2009, 02:00 PM
Salamandar
Inverse With Restricted Domain
I need to find the domain and range of the inverse of this function:

$f(x)=x^3+2$

I ended up with $f^{-1}(x)=\sqrt[3]{x-2}$ as the inverse . . I think . . . All the examples I have come across with inverses do not have a cube root, only square root.

(Crying) So confused.
• Nov 18th 2009, 02:51 PM
JSB1917
$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

So you should be able to plug both of those into the corresponding equations and they should equal x. This is helpful especially if you are given two equations and asked if one is the inverse of the other.

In order to get inverse functions you can solve for x in the original equation, then exchange x and y, or interchange x and y and then solve. (Whoops, sorry about that, edited it)

What's the cube root of -64 and what's the cube root of -27? Hint, hint.
• Nov 19th 2009, 10:28 AM
Hello Salamandar
Quote:

Originally Posted by Salamandar
I need to find the domain and range of the inverse of this function:

$f(x)=x^3+2$

I ended up with $f^{-1}(x)=\sqrt[3]{x-2}$ as the inverse . . I think . . . All the examples I have come across with inverses do not have a cube root, only square root.

(Crying) So confused.

See the attached sketch of the function.

You'll see that for each value of $y$ there's one and only one value of $x$, so the inverse function has the whole of $\mathbb{R}$ as its domain and its range.