Every invertible function is

A function is invertible when it is both one-one and onto

For one-one function, for every $y$, there should only be one value of $x$. This is possible, when function the graph of $y=f\left(x\right)$ should be monotonic.

If we consider the cos of parabola, where it touches the' $x$-axis twice,

For $y=0$
For one-one junction, for every $y,$ there should be only one value of $x$. For this the function should either be cither increasing or decreasing only.

If we consider the case of a hara bola, $y=x^2$ which decreases in $(-\infty ,0)$ and increases in $(0,\infty )$, if we draw a line $y=k$, it touches curve at $2$ points. $\Rightarrow y$ has two values of '$x$'. Therefore jor function to be invertible it has to be monotonic.