Inverse Functions and Graphs

If $f : A \to B$ is bijective (Section 2), then for every $b \in B$ there is exactly one $a \in A$ with $f(a)=b$. This lets us define a new function, the inverse function $f^{-1} : B \to A$, by

In effect $f^{-1}$ reverses every ordered pair of $f$: if $(a,b)\in f$, then $(b,a)\in f^{-1}$. An inverse "undoes" what the original function does, in the precise sense that

Only a bijective function is guaranteed to have an inverse that is itself a function — this is exactly why bijectivity was singled out in Section 2. (A function that is one-one but not onto can be made invertible by restricting its codomain to its range.)

To find the inverse of a function given by a formula $y=f(x)$:

1. Write $y=f(x)$.
2. Solve this equation algebraically for $x$ in terms of $y$.
3. Interchange the roles of the variables, writing $f^{-1}(x)$ for the resulting expression (relabelling $y$ as $x$).
4. State the domain of $f^{-1}$, which is the range of $f$.

Graphs and inverses. The graph of $f^{-1}$ is always the mirror image of the graph of $f$ in the line $y=x$ — reflecting each point $(a,b)$ to $(b,a)$. This is why the exponential graph $y=a^x$ and the logarithmic graph $y=\log_a x$ of the same base are reflections of one another.

…