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