# Inverse Functions

**Recall:** The inverse of a function f is a function itself if and only
if f is one-to-one.

Let f be a one-to-one function. Then g is the
inverse function of
f if |

If g is the inverse function of f, then we write g as f^{-1}(x) and read:
“f-inverse”.

**Example:** Determine whether the following functions
are inverses of each other:

## Cancellation Rules for Inverses

The inverse functions undo each other with
respect to the compositions: f |

** Equivalent Form of the Cancellation Rules:**

f (x) = y | <-> | f^{-1}( y) = x |

(x in domain of f) | (y in the domain of f^{-1}) |

**Note on the Domains and Ranges of the Inverses:**

Domain of f^{-1} = Range of fRange of f ^{-1} = Domain of f |

## Graphing Inverses :

If the graph of f is the set of points (x, y), then the
graph

of f^{-1}is the set of points ( y, x).

Since, points (x, y) and ( y, x) are symmetric with

respect to the line y = x

then

the graphs of f and f ^{-1} are symmetric with
respect tothe line y = x. |

**Example: **Given the graph of y = f (x). Draw the
graph of its inverse.

## Finding the Inverse of a One-to-one Function f:

1. Write y = f (x).

2. Solve the equation for x : x = f^{-1} ( y)

3. Interchange x and y.

4. Give your answer in terms of f ^{-1} (x).

**Note:** Consider all restrictions on the variables.

**Example: **Find f^{-1}(x), if it is possible.

**Finding the Inverse of a Domain-restricted Function:**

(a) Find the inverse of

(b) The function

is one-to-one.

Find f^{-1}(x) and its domain and range.

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