Inverse of a function not defined vs function not invertible

Question

So let's say that we have a function $f(x) = x^2$ (just to give an example).

Is there a difference between saying that f(x) is not invertible and saying that $f^{-1}(x)$ is not defined?

To me, it sounds like these are two different ways of saying that $f(x)$ has no inverse, but something tells me I am wrong.

Answer 1

Saying that $f$ is not invertible means that the function $f^{-1}$ does not exist. Saying that $f^{-1}(x)$ is not defined means that the value of $f^{-1}(x)$ does not exist. These are slightly different things.

For example, let $f:[0,\infty)\to\mathbb R$ be defined by $x\mapsto x$ (the identity). This function is certainly invertible, but for example $f^{-1}(-1)$ is not defined, since no $x\in [0,\infty)$ exists that makes $f(x)=-1$.

Of course, it is quite common to be sloppy and write $f^{-1}(x)$ when one really means the function $f^{-1}$. So it is possible that you would see something like "if $f:\mathbb R\to\mathbb R$ is defined by $f(x)=x^2$, then $f^{-1}(x)$ is not defined", when what one really means is that the function $f^{-1}$ is not well-defined. Usually, you can easily tell what is meant from context, so this isn't a major issue.

Answer 2

We say that $f(x)=x^2$ as function $\mathbb R \to \mathbb R$ is not invertible or also that the inverse is not defined but recall that $f(x)=x^2$ as function $\mathbb R^+\cup\{0\} \to \mathbb R^+\cup\{0\}$ is invertible, that is the inverse $g(x)=\sqrt x$ is defined.