what makes a function invertible?

Question

From what I understand invertible is if we can equate x = f(x) of some form.. Whereas Inverse is where the function can be reflected across the y=x axis. Are those related in any way?

Is it true that only $f(x)=\frac{1}{x}$ out of these below functions are invertible? If so, why?

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$f(x)=x^2, f(x)=|x|, f(x)=\frac{1}{x}, f(x)=2$

Answer 1

A function is invertible if and only if it is one-to-one. A one-to-one function is a function where no two inputs produce the same output, i.e. for all $a$ and $b$ in the domain of $f$, $$ f(a)=f(b)\implies a=b \, , $$ or, equivalently, $$ a\neq b\implies f(a)\neq f(b) \, . $$

As Martin R mentions in the comments, if $f(x)=x$ for some $x$, then $f$ is said to have a "fixed point" at $x$. This has nothing to do with whether $f$ is invertible.

So for each of your functions, you have to consider whether $f(a)=f(b)$ implies $a=b$. For instance, does $a^2=b^2$ imply that $a=b$?

Answer 2

$f$ is invertible (or injective) if whenever $f(x)=f(y)$, we have $x=y$. Or equivalently for each $z$ in the range of $f$ there is only one $x$ such that $f(x)=z$. For $1,2$ and $4$ you should try to find $f(x)=f(y)$ where $x\neq y$.

Hint: For $1$ we have $1^2=(-1)^2$.