Right Inverse of a Function Explained

[jgens]

Could someone please explain what is implied if a function has a right inverse? Thanks.

 

[NoMoreExams]

Have you read this: http://en.wikipedia.org/wiki/Inverse_function#Left_and_right_inverses

 

[Tac-Tics]

Let f be a function.

If r is the right inverse of f, then for all x, f(r(x)) = x. That is, the composition of f and r, f * r, is the identity function.

If l is a left inverse of f, then for all x, l(f(x)) = x. Again, this means l * f is the identity function.

If a function g is both a left and a right inverse, it is called a full inverse (or just simple, THE inverse). The full inverse of of f is usually designated f^-1.

Some examples:

The squaring function, f(x) = x^2, is not one-to-one, and so it has no full inverse. However, it does have a partial inverse (a left inverse) which is the square root function. We know this because sqrt(x^2) = x. We can show it is not a full inverse by demonstrating that for some x, (sqrt(x))^2 /= x, and we can let x be any negative number. (Note in the complex numbers, sqrt is in fact a full inverse).

 

[jgens]

NoMoreExams: Thanks, I had not read that article. That clears a lot of things up.

Tac-Tics: Thanks for the example.