inverse function

in mathematics, inverse refers to either the multiplicative inverse, also known as reciprocal, or an inverse function

for function $f$, the function that makes the output be the input would be $f^{-1}$
for example, if $f=a^{\frac{b}{c}}$ then $f^{-1}=a^{\frac{c}{b}}$

if the original function is a one-to-one relation, the inverse function should be able to reverse everything correctly, and $f^{-1}(f(x))=x$
for example, if $f(x)=x+5$, then $f^{-1}(x)=x-5$

if the original function is a many-to-one relation, then there would be no possible inverse for the function.
for example, if $f(x)=x^2$ then we have a inverse function that can input $f(x)$ and return what $x$ was originally.
however, if we set a constraint and make the function $f(x)=x^2,x\ge 0$ it becomes a one-to-one relation, so the inverse would be $f^{-1}(x)=\sqrt{x},x\ge 0$

you might realize, you can always get the inverse function by swapping the input and output, like $f:f(x)=x^2$ to $f^{-1}:x=f^{-1}(x{)}^2$
also, the inverse function looks like the original function reflected across the line $y=x$ because the axes are swapped.