Inverse of a Given Function
Big Idea:
An inverse function "undoes" what the original function does. If $f$ takes $x$ to $y$, then the inverse function $f^{-1}$ takes $y$ back to $x$.
Formal Definition
A function $f$ with domain $X$ and range $Y$ is called invertible if the inverse correspondence between $Y$ and $X$ is also a function.
If $f$ is invertible, the inverse correspondence is called the inverse function of $f$, denoted $f^{-1}$.
Key Notation: $f^{-1}(x)$ does not mean $\frac{1}{f(x)}$. The superscript $-1$ here denotes the inverse function, not a reciprocal!
Mathematically, if $f(a) = b$, then $f^{-1}(b) = a$. We can write:
$$f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x$$