If I want to state if a function is invertible I have couple of tools.
I can check if the function is odd or even.
If it is even, it is certainly not invertible, since $f(x)=f(-x)$, it is strictly agains definition of invertible function. If it is odd, it does not mean it is invertible, because $sin(x)$ is not invertible(periodic) and odd compound function $\frac{x^3}{x^2-1}$ is also not invertible on all real numbers(without restriction), because there are $x_1 \neq x_2$, but $f(x_1)=f(x_2)$ in some cases.
So if I do not want to calculate invertibility of a function with definition, I should go for derivation. If derivation exists on some inside interval of function, I can check stationary points and if they do not represent extreme(for example $x^3$) and is all plus or minus, I can state that this function is on that interval invertible, am I right?