# Inverse of a Function

Recall that a function between two variables, x and y, describes the pattern of dependence of one variable, say y, in terms of the other variable x. This pattern tells us how x is transformed to become y. An inverse function, if it exists, inverts this pattern and tells us how y must be transformed to recover x. For example,

- Subtraction is the inverse of addition and vice-versa
- Division is the inverse of multiplication and vice-versa
- Square-root is the inverse of squaring and vice-versa
- Etc.

If y = f(x) is a function of x, then we know that to each given x value there is exactly one y value. This is verified graphically by the **vertical line test**. For the existence of the inverse function of y = f(x), it must be also true that for each given y there must be exactly one x value. This property is verified by the horizontal line test. If a function * f* passes both vertical and horizontal line tests then

*is called a*

**f****one-to-one function**. It is a fact that inverse functions exist for one-to-one functions. For example, the linear function

*, m ≠ 0, is a one-to-one function and its inverse is*

**y = mx + b***.*

**x = (y – b)/m**Not all functions are one-to-one. For example, **y = f(x) = x**** ^{2}**,

*, is not one-to-one since the value*

**-∞ < x < ∞***is same for both*

**y***and*

**x***. However, by suitable restriction of the domain a function can be made into a one-to-one function. For example, restricting x to the set of non-negative numbers,*

**-x**

**y = f(x) = x****becomes one-to-one. The inverse of this function is**

^{2}*\(x = f^{-1}(y) = \sqrt{y}\)**, *

where y ≥ 0.

**Review: **Composition of functions, function arithmetic, completing-the-square method for quadratic functions, solving rational equations.

**Learning Objectives: **In this module we will learn

**One-to-one function**: verify if a function f is one-to-one both graphically and algebraically.- Determine the domain and range of a one-to-one function f and its inverse f
^{-1}. - Algebraically determine the inverse of a given one-to-one function.
- Algebraically verify the properties of an inverse function
- Determine the graph of inverse function f
^{-1}^{ }from the graph of f. - Apply inverse functions.