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Every one-to-one function *f* has an
inverse function *f*^{-1} which essentially reverses the operations performed by
*f*.

More formally, if *f* is a one-to-one function with domain *D* and range *R*, then its inverse *f*^{-1}
has domain *R* and range *D*. *f*^{-1} is related to *f* in the following way:
If *f* (*x*) = *y*, then *f*^{-1}(*y*) = *x*.
Written another way, *f*^{-1}(*f* (*x*)) = *x*.

Example: *f* (*x*) = 3*x* - 4. Find *f*^{-1}(*x*).

The procedure for finding *f*^{-1}(*x*) from *f* (*x*) involves first solving for *x* in terms
of *y*.

y | = 3x - 4 | ||

x | = |

Now switch the variables *x* and *y* in the equation to generate the inverse:

y | = | ||

f^{-1}(x) | = |

A function and its inverse are related geometrically in that they are reflections about the
line *y* = *x*:

Figure %: A function and its inverse are symmetric with respect to the line *y* = *x*

Thus, if (*a*, *b*) is a point on the graph of *f*, then (*b*, *a*) is a point on the graph of *f*^{-1}.

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