# Inverse functions

## Understanding Inverse Functions

An inverse function is the opposite of the original function, represented by **f ^{-1}(x)**, while the original function is

**f(x)**.

Inverse functions exist only for one-to-one functions, where one input value leads to a single output value. This is unlike one-to-many functions, where one input value can yield multiple output values.

## How to Find the Inverse of a Function

There are three simple steps to finding the inverse of a function:

- Substitute
**y**for the function notation (e.g.**f(x)**becomes**y**). - Rearrange the function to make
**x**the subject. - Replace
**x**with the inverse function notation and**y**with**x**(e.g.**x**becomes**f**).^{-1}(x)

For example, to find the inverse function of **f(x) = 3x - 2**:

- Substitute
**y**:**y = 3x - 2**. - Rearrange to make
**x**the subject:**x = (y+2)/3**. - Replace
**x**and**y**with inverse function notation:**f**.^{-1}(x) = (x+2)/3

## Solving Inverse Function Problems

Inverse functions can be used to solve different types of questions:

- When
**x**is given: substitute the value of**x**into the inverse function and solve. - When the function is set to a value: set the inverse function equal to
**y**, rearrange the equation to isolate**x**, and solve for**x**. - For domains and ranges: the domain of the inverse function is the range of the original function, and vice versa.

## Graphical Representation of Inverse Functions

There are two ways to graphically represent inverse functions:

- Directly reflect the original function in the line
**y = x**, using transformation skills for graphs. - Find the inverse function and plot the coordinates of
**x**and**y**on a graph.

To directly reflect the original function in the line **y = x**, use the original function and the line as the line of reflection. For instance, **f(x) = 2x + 5** can be graphed as:

- The original function (red).
- The original function (red) and line
**y = x**(blue) as the line of reflection. - The inverse function (green) obtained by reflecting the original function (red) in the line of reflection (blue).

If the original function involves a variable raised to a power other than 1, the process may be more complicated.

## Inverse Functions - Key Points

- Inverse functions are the opposite of the original function.
- Their notation differs from typical functions, using
**f**.^{-1}(x) - Only one-to-one functions have inverse functions.
- Inverse functions can be found by substituting
**y**for the function notation, rearranging the equation, and replacing variables with inverse function notation. - The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.