Subject:
Mathematics
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
Mathematics, Tmm0022
Language:
English
Media Formats:
Text/HTML

# Section 1: Inverse of a Function

Emphasize the relationship between the domain and range of a one-to-one function and its inverse. Make sure that students understand the difference between y and x as notated in y = f(x) and y = f-1(x).

Finding inverses of quadratic and rational functions is a good exercise.

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 f is called a one-to-one function. It is a fact that inverse functions exist for one-to-one functions. For example, the linear function y = mx + b, m ≠ 0, is a one-to-one function and its inverse is x = (y – b)/m.

Not all functions are one-to-one. For example, y = f(x) = x2, -∞ < x  < ∞, is not one-to-one since the value y is same for both x 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, y = f(x) = x2 becomes one-to-one. The inverse of this function is

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

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.