Inverse Functions and One-to-One Functions: Study Notes for College Algebra

Inverse Functions

Introduction to Inverse Functions

Inverse functions are a fundamental concept in algebra, allowing us to reverse the process of a function to recover its input from its output. Understanding when a function has an inverse and how to find it is essential for solving equations and modeling real-world scenarios.

• Inverse Relation: Interchanging the first and second coordinates of each ordered pair in a relation produces the inverse relation.

• Inverse Function: If the inverse relation is also a function, it is called an inverse function.

• Notation: The inverse of a function f is denoted by f-1, read as "f-inverse." The -1 is not an exponent.

Finding the Inverse of a Relation

To find the inverse of a relation, swap the x and y values in each ordered pair.

• Example: If h = {(-8, 5), (4, -2), (-7, 1), (3.8, 6.2)}, then the inverse is {(5, -8), (-2, 4), (1, -7), (6.2, 3.8)}.

Inverse Relation from an Equation

If a relation is defined by an equation, interchanging the variables produces an equation of the inverse relation.

• Example: Given y = x^2 - 2x, the inverse equation is x = y^2 - 2y.

Graphs of a Relation and Its Inverse

The graph of a relation and its inverse are always reflections of each other across the line y = x.

• Key Point: Interchanging x and y in the equation or ordered pairs reflects the graph over the line y = x.

• Example: For g = {(2, 4), (-1, 3), (-2, 0)}, the inverse is {(4, 2), (3, -1), (0, -2)}.

One-to-One Functions

Definition and Properties

A function is one-to-one if each output is produced by exactly one input. This property is crucial for a function to have an inverse that is also a function.

• Formal Definition: If a ≠ b, then f(a) ≠ f(b). Alternatively, if f(a) = f(b), then a = b.

• Increasing/Decreasing Functions: A function that is strictly increasing or decreasing over its domain is one-to-one.

Horizontal-Line Test

The Horizontal-Line Test is a graphical method to determine if a function is one-to-one.

• If any horizontal line intersects the graph of a function more than once, the function is not one-to-one, and its inverse is not a function.

• If every horizontal line intersects the graph at most once, the function is one-to-one, and its inverse is a function.

Examples

• Example 1: f(x) = 4 - x is one-to-one (passes the horizontal-line test).

• Example 2: f(x) = x^2 is not one-to-one (fails the horizontal-line test).

• Example 3: f(x) = \sqrt[3]{x + 2} + 3 is one-to-one.

• Example 4: f(x) = 3x^5 - 20x^3 is not one-to-one.

Obtaining a Formula for an Inverse Function

Step-by-Step Process

If a function f is one-to-one, its inverse can be found using the following steps:

1. Replace f(x) with y.

2. Interchange x and y.

3. Solve for y.

4. Replace y with f-1(x).

• Example: For f(x) = 2x - 3:

  1. Let y = 2x - 3

  2. Interchange: x = 2y - 3

  3. Solve: x + 3 = 2y ⇒ y = \frac{x + 3}{2}

  4. So, f-1(x) = \frac{x + 3}{2}

Graphical Comparison

The graph of f-1 is a reflection of the graph of f across the line y = x.

Table: Comparison of Function and Its Inverse

x	f(x) = 2x - 3	f-1(x) = \frac{x + 3}{2}
-1	-5	-1
0	-3	0
1	-1	1
2	1	2
3	3	3

Inverse Functions and Composition

Properties of Inverse Functions

If f is one-to-one, then f-1 is the unique function such that:

• $f^{-1}(f(x)) = x$ for each x in the domain of f

• $f(f^{-1}(x)) = x$ for each x in the domain of f^{-1}

• Example: Given f(x) = 5x + 8, then f^{-1}(x) = \frac{x - 8}{5}.

  • $f^{-1}(f(x)) = f^{-1}(5x + 8) = \frac{5x + 8 - 8}{5} = x$

  • $f(f^{-1}(x)) = 5 \left( \frac{x - 8}{5} \right) + 8 = x - 8 + 8 = x$

Restricting the Domain

Why Restrict the Domain?

Sometimes, the inverse of a function is not a function unless the domain is restricted. This is common with functions like f(x) = x^2, which are not one-to-one over all real numbers.

• Example: For f(x) = x^2, the inverse relation is x = y^2, or y = ±\sqrt{x}, which is not a function.

• Restriction: If we restrict the domain to x ≥ 0, then the inverse is f^{-1}(x) = \sqrt{x}.

Table: Domain Restriction Example

Function	Domain	Inverse
f(x) = x^2	All real numbers	Not a function
f(x) = x^2	x ≥ 0	f^{-1}(x) = \sqrt{x}

Summary of Key Concepts

• Inverse functions reverse the effect of the original function.

• A function must be one-to-one to have an inverse that is also a function.

• The horizontal-line test is used to determine if a function is one-to-one.

• To find an inverse, swap variables and solve for the new output.

• Sometimes, domain restriction is necessary for the inverse to be a function.

• Inverse functions satisfy $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$.