Let's assume we have a function $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = x^5$. Then the inverse function would be $f^{-1}=\sqrt[5]{x}$ and $f^{-1} \circ f = f \circ f^{-1} = e$ is the identity function. Does finding this inverse function suffice to prove that $f$ is bijective or do we need to prove injectivity and surjectivity for $f$ seperately?
Does finding an inverse function prove it’s bijective
elementary-set-theoryfunctionsinverse function
Best Answer
Yes. A function $f$ has an inverse function $f^{-1}$ (a function such that $f \circ f^{-1} = f^{-1} \circ f = \text{id}$ for your identity function $\text{id}$) if and only if $f$ is a bijection. You may see a proof of this here.
Of course, be sure to demonstrate that the composition does hold - in your case you need to show $(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$, but this is almost trivial.