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Here it is:
 

A function, f(x), has an inverse function
if
f(x) is one-to-one.

I know what you're thinking:

"Oh, yeah!  Thanks a heap, math geek lady.  That's very helpful!"

Come on!  You know I'm going to tell you what one-to-one is!  Have I let you down yet?

OK, one-to-one...

There's an easy way to look at it, then there's a more technical way.  (The technical way will really get us off track, so I'm leaving it out for now.)

Here's the easy way:

The Horizontal Line Test:

If you can draw a horizontal line so that it hits the graph in more than one spot, then it is NOT one-to-one.

Check it out:

Isf( x ) = x^( 3 )one-to-one?
a graph of f( x ) = x^( 3 ) ... it passes the horizontal line testYep!
So,f( x ) = x^( 3 )HAS an
inverse function.