Inverse Functions and One-to-One Functions (Section 4.1 Study Notes)

Inverse Functions and One-to-One Functions

Introduction

This study guide covers the key concepts from Section 4.1 on Inverse Functions in a Precalculus course. It includes definitions, properties, graphical interpretations, and step-by-step procedures for finding inverses, as well as illustrative examples and applications.

One-to-One Functions

Definition and Properties

• Function: A relation where each input x corresponds to exactly one output y.

• One-to-One Function: A function is one-to-one if each output y corresponds to only one input x.

• Horizontal Line Test: A function is one-to-one if every horizontal line intersects its graph at most once.

Example: The function $f(x) = x^2$ is not one-to-one on $(-\infty, \infty)$ because, for example, $f(2) = f(-2) = 4$. However, if the domain is restricted to $x \geq 0$, it becomes one-to-one.

Inverse Functions

Definition and Notation

• The inverse of a function $f$ is denoted $f^{-1}$.

• If $f$ maps $x$ to $y$ ($f(x) = y$), then $f^{-1}$ maps $y$ back to $x$ ($f^{-1}(y) = x$).

• For $f$ and $f^{-1}$ to be inverses: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in their respective domains.

Example (Process Diagram):

• Input $x$ → $f$ → Output $f(x)$

• Input $f(x)$ → $f^{-1}$ → Output $x$

Graphical Interpretation

• The graph of $f^{-1}$ is the reflection of the graph of $f$ over the line $y = x$.

• The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

Finding the Inverse of a Function

Step-by-Step Procedure

1. Replace $f(x)$ with $y$ in the equation defining $f(x)$.

2. Interchange $x$ and $y$.

3. Solve the new equation for $y$.

4. Write $f^{-1}(x)$ for $y$.

Examples

• Example 1: $f(x) = x + 5$ $f^{-1}(x) = x - 5$

• Example 2: $f(x) = 2x$ $f^{-1}(x) = \frac{x}{2}$

• Example 3: $f(x) = \sqrt{x}$ $f^{-1}(x) = x^2$ (with domain $x \geq 0$)

• Example 4: $f(x) = 2x - 3$ $f^{-1}(x) = \frac{x + 3}{2}$

• Example 5: $f(x) = \frac{2x - 1}{3x + 2}$ To find the inverse, follow the step-by-step procedure: 1. $y = \frac{2x - 1}{3x + 2}$ 2. Interchange $x$ and $y$: $x = \frac{2y - 1}{3y + 2}$ 3. Solve for $y$: $x(3y + 2) = 2y - 1$ $3xy + 2x = 2y - 1$ $3xy - 2y = -2x - 1$ $y(3x - 2) = -2x - 1$ $y = \frac{-2x - 1}{3x - 2}$ 4. $f^{-1}(x) = \frac{-2x - 1}{3x - 2}$

Verifying Inverses

Algebraic Verification

• Functions $f(x)$ and $g(x)$ are inverses if: $f(g(x)) = x$ and $g(f(x)) = x$

• Example: $f(x) = 8x - 2$, $g(x) = \frac{1}{8}x + \frac{1}{4}$ Check: $f(g(x)) = 8\left(\frac{1}{8}x + \frac{1}{4}\right) - 2 = x + 2 - 2 = x$ $g(f(x)) = \frac{1}{8}(8x - 2) + \frac{1}{4} = x - \frac{1}{4} + \frac{1}{4} = x$

Domain and Range of Inverse Functions

• The domain of $f^{-1}$ is the range of $f$.

• The range of $f^{-1}$ is the domain of $f$.

Applications of Inverse Functions

Example: Pharmacokinetics

• Suppose a patient has alcohol in their system, and the concentration $C$ in the blood (gm/mL) is a function of time $t$ in hours: $C = f(t)$.

• If you want to know how long until the concentration reaches a certain value (e.g., $C = 3$), you need the inverse function: $t = f^{-1}(C)$.

• This is a practical example of inverting a function to solve for the input given an output.

Graphical Example: Inverse of a Quadratic Function

• Example: $f(x) = (x - 3)^2$, defined on $x \geq 3$

• To find the inverse, solve $y = (x - 3)^2$ for $x$:

$y = (x - 3)^2$ $\sqrt{y} = x - 3$ (since $x \geq 3$, take the positive root) $x = \sqrt{y} + 3$ Therefore, $f^{-1}(x) = \sqrt{x} + 3$ (with domain $x \geq 0$)

Summary Table: Properties of Functions and Their Inverses

Property	Function $f$	Inverse $f^{-1}$
Domain	$D$	Range of $f$
Range	Range of $f$	$D$
Graph	Original function	Reflection over $y = x$
One-to-one?	Required for inverse to exist	Always a function if $f$ is one-to-one

Practice Problems

• Find the inverse of $f(x) = \frac{2}{x + 5}$.

• Find the inverse of $f(x) = 2x - 3$.

• Verify that $f(x) = 8x - 2$ and $g(x) = \frac{1}{8}x + \frac{1}{4}$ are inverses.

• Given $f(x) = (x - 3)^2$ for $x \geq 3$, find $f^{-1}(x)$ and state its domain.

Additional info: The notes also include exam logistics and reminders, but the mathematical content is focused on inverse functions, one-to-one functions, and their properties and applications.