Inverse Functions

If $f$ is a function, the inverse function $f^{-1}$ (if it exists) undoes whatever $f$ does. That is,

$y=f^{-1}(x)\quad \Longleftrightarrow \quad x=f(y)$ and $x=f^{-1}(y) \quad \Longleftrightarrow \quad y=f(x)$

A few facts:

• $f(f^{-1}(\Box))=\Box$, no matter what you put into the box.
• The domain of $f^{-1}$ is the range of $f$; and the range of $f^{-1}$ is the domain of $f$. So, the independent variable of $f$ is now the dependent variable of $f^{-1}$ and vice versa. Graphically, this means that the horizontal axis becomes vertical, and the vertical axis horizontal.
• The function $f^{-1}$ exists only if $f$ is one-to-one. Graphically, $f^{-1}$ is a function if its graph passes the vertical line test, which means that the graph of $f$ must pass the horizontal line test. If $f$ is not one-to-one, we must restrict the domain of $f$ first.

For instance, if $f(x)=x+1$, then the inverse function is $f^{-1}(x)=x-1$, since subtracting 1 undoes the effect of adding 1. If $f(x)=2x$, then $f^{-1}(x)=x/2$, since dividing by 2 undoes the effect of multiplying by 2. We'll explore how to find the formula for $f^{-1}(x)$ in general on the next slide.