One-to-One Functions and Their Inverses: College Algebra Study Notes

One-to-One Functions and Their Inverses

Introduction

This study guide covers the concept of one-to-one functions, their properties, methods for determining if a function is one-to-one, and the process of finding and interpreting inverse functions. These topics are foundational in College Algebra and are essential for understanding function behavior and solving equations involving inverses.

One-to-One Functions

Definition and Properties

• One-to-one function: A function f with domain A is called one-to-one (injective) if no two different elements in the domain map to the same image in the codomain.

• Formally, $f(x_1) eq f(x_2)$ whenever $x_1 eq x_2$ Alternatively, if $f(x_1) = f(x_2)$, then $x_1 = x_2$.

• One-to-one functions are important because only they have inverses that are also functions.

Arrow Diagram Example

Consider two functions f and g represented by arrow diagrams:

• f is one-to-one: Each element in the domain maps to a unique element in the codomain.

• g is not one-to-one: Two or more elements in the domain map to the same element in the codomain.

Horizontal Line Test

• Geometric method: A function is one-to-one if and only if no horizontal line intersects its graph more than once.

• This is known as the Horizontal Line Test.

Example: Deciding Whether a Function Is One-to-One

• Consider $f(x) = x^3$.

• For any $x_1 eq x_2$, $x_1^3 eq x_2^3$ (since the cubic function is strictly increasing).

• By the Horizontal Line Test, the graph of $f(x) = x^3$ is intersected at most once by any horizontal line, so $f$ is one-to-one.

Monotonic Functions

• Increasing or decreasing functions: Every strictly increasing or strictly decreasing function is one-to-one.

Inverse of a Function

Definition and Notation

• The inverse function of f, denoted $f^{-1}$, reverses the effect of f.

• If $f(a) = b$, then $f^{-1}(b) = a$.

• Important: $f^{-1}(x)$ does not mean $1/f(x)$; the notation refers to the inverse function, not the reciprocal.

Inverse Function Properties

• If f is a one-to-one function with domain A and range B, then its inverse $f^{-1}$ has the following properties:

• $f^{-1}(f(x)) = x$ for every $x$ in $A$

• $f(f^{-1}(x)) = x$ for every $x$ in $B$

Finding the Inverse of a Function

• To find the inverse of a function, follow these steps:

  1. Replace $f(x)$ with $y$.

  2. Solve the equation for $x$ in terms of $y$.

  3. Interchange $x$ and $y$ to write the inverse function $f^{-1}(x)$.

Example: Linear Function

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

• Let $y = 3x - 2$

• Add 2: $y + 2 = 3x$

• Divide by 3: $x = rac{y + 2}{3}$

• Interchange $x$ and $y$: $f^{-1}(x) = rac{x + 2}{3}$

Example: Rational Function

• Find the inverse of $f(x) = rac{2x + 3}{x - 1}$.

• Let $y = rac{2x + 3}{x - 1}$

• Multiply both sides by $(x - 1)$: $y(x - 1) = 2x + 3$

• Expand: $yx - y = 2x + 3$

• Bring $x$ terms together: $yx - 2x = y + 3$

• Factor $x$: $x(y - 2) = y + 3$

• Divide: $x = rac{y + 3}{y - 2}$

• Interchange $x$ and $y$: $f^{-1}(x) = rac{x + 3}{x - 2}$

Graphing the Inverse of a Function

Reflection Across the Line $y = x$

• The graph of $f^{-1}$ is obtained by reflecting the graph of $f$ across the line $y = x$.

• If $(a, b)$ is a point on the graph of $f$, then $(b, a)$ is a point on the graph of $f^{-1}$.

Example: Square Root Function

• Given $f(x) = ext{sqrt}(x - 2)$, to find $f^{-1}$:

• Let $y = ext{sqrt}(x - 2)$

• Square both sides: $y^2 = x - 2$

• Solve for $x$: $x = y^2 + 2$

• Interchange $x$ and $y$: $f^{-1}(x) = x^2 + 2$

Applications of Inverse Functions

Real-World Modeling

• Inverse functions are used to reverse processes in real-world situations.

• Variables are often chosen to reflect the context (e.g., $t$ for time, $d$ for distance, $V$ for volume).

• If $R = f(N)$ models a relationship, then $f^{-1}$ gives $N$ as a function of $R$.

Example: Pizza Pricing

• A pizza parlor charges $12 for a plain cheese pizza plus $2 for each additional topping.

• Let $f(n) = 12 + 2n$ be the price $p$ as a function of the number of toppings $n$.

• To find the inverse, solve $p = 12 + 2n$ for $n$:

• $p - 12 = 2n$

• $n = rac{p - 12}{2}$

• Thus, $f^{-1}(p) = rac{p - 12}{2}$ gives the number of toppings for a given price.

• For a pizza costing $22$,$f^{-1}(22) = rac{22 - 12}{2} = 5$ toppings.

Summary Table: Properties of One-to-One and Inverse Functions

Concept	Definition	Key Property
One-to-One Function	Each input maps to a unique output	$f(x_1) eq f(x_2)$ for $x_1 eq x_2$
Inverse Function	Reverses the effect of the original function	$f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$
Horizontal Line Test	Geometric test for one-to-one	No horizontal line intersects the graph more than once
Graph of Inverse	Reflection across $y = x$	If $(a, b)$ on $f$, then $(b, a)$ on $f^{-1}$

Additional info: The notes have been expanded with formal definitions, step-by-step examples, and a summary table for clarity and completeness.