Question

For the function f(x) = x2 ­ -12, find (f o f­-1)(4) A. 0 B. 4 C. 10 D. 6

Answer 1

We actually don't need to do any computation. By definition, the inverse function $f^{-1}$ changes the role of input and output. So, if a function f maps x onto y, the inverse function maps y onto x.

You have to think like this: if the function makes a step further, the inverse function makes that same step back.

This means that the composition $f \circ f^{-1}$ is always the identity function $f(x) = x$. In fact,

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

So, for every function, you have

$(f\circ f^{-1})(4) = f(f^{-1}(4)) = 4$