One-to-One Functions

MATH162

Reference: Stewart 6.1 • Chapter: 6 • Section: 1

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One-to-One Functions

Why One-to-One Matters

Not every function has an inverse. Consider the function $f(x) = x^2$: both $f(2) = 4$ and $f(-2) = 4$. If we tried to "reverse" $f$ and ask "what input gives output 4?", there is no unique answer—it could be $2$ or $-2$.

Functions that do have inverses are called one-to-one functions. Recognizing them is the gateway to the entire theory of inverse functions, logarithms, and inverse trigonometric functions.

Before You Start: Quick Self-Check

Can you answer these prerequisite questions?

What is a function? A rule that assigns exactly one output to each input.
What does $f(3) = 7$ mean? When the input is 3, the output is 7.
Can you sketch the graph of $y = x^2$? A parabola opening upward.

If you struggled with these, review first

These concepts are essential prerequisites. Review the Function Definition skill before continuing.

Prerequisite Map

Quick Reference

Property	Value
Concept	Inverse Functions
Chapter	6.1
Difficulty	Beginner
Time	~15 minutes

Key Concepts

Definition

$$\boxed{\text{A function } f \text{ is \textbf{one-to-one} if } f(x_1) \neq f(x_2) \text{ whenever } x_1 \neq x_2}$$

In other words: different inputs always produce different outputs.

Equivalently: if $f(x_1) = f(x_2)$, then $x_1 = x_2$.

The Horizontal Line Test

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

     y                              y
     |    ___                       |     /
     |   /   \                      |    /
     |  /     \                     |   /
-----|--+------\---- x          ----+--/--------- x
     |          \                   | /
     |           \_                 |/
     |                              |
   NOT one-to-one               One-to-one
   (line crosses twice)         (each line crosses once)

Why It Works

If a horizontal line $y = k$ intersects the graph at two points $(x_1, k)$ and $(x_2, k)$, then $f(x_1) = k = f(x_2)$ with $x_1 \neq x_2$. This violates the one-to-one property.

Common Examples

Function	One-to-One?	Reason
$f(x) = x^3$	Yes	Strictly increasing; passes HLT
$f(x) = x^2$	No	$f(2) = f(-2) = 4$
$f(x) = x^2, \, x \geq 0$	Yes	Restricted domain makes it one-to-one
$f(x) = 2x + 5$	Yes	Linear with nonzero slope
$f(x) = \sin x$	No	Periodic; fails HLT
$f(x) = e^x$	Yes	Strictly increasing

Watch Out For: Edge Cases

Situation	What Happens	Example
Constant functions	Never one-to-one	$f(x) = 5$ gives same output for all inputs
Even functions	Not one-to-one on symmetric domains	$f(x) = x^4$ has $f(a) = f(-a)$
Periodic functions	Not one-to-one unless domain restricted	$\sin x$, $\cos x$ repeat values
Piecewise functions	Check each piece AND transitions	Must verify at boundaries too

Algebraic Test

To prove a function is one-to-one algebraically:

Assume $f(x_1) = f(x_2)$
Show this forces $x_1 = x_2$

Example: Prove $f(x) = 5x - 2$ is one-to-one.

If $f(x_1) = f(x_2)$, then: $$5x_1 - 2 = 5x_2 - 2$$ $$5x_1 = 5x_2$$ $$x_1 = x_2$$ ✓

Connection to Increasing/Decreasing Functions

Theorem: If $f$ is strictly increasing on its domain, then $f$ is one-to-one.

Similarly for strictly decreasing functions.

Why? If $x_1 < x_2$, then $f(x_1) < f(x_2)$ (or $>$ for decreasing), so $f(x_1) \neq f(x_2)$.

This gives a calculus criterion: if $f'(x) > 0$ everywhere (or $f'(x) < 0$ everywhere), then $f$ is one-to-one.

Practice Problems

Level 1 Identifying from a Table

Is the function given by this table one-to-one?

$x$	1	2	3	4	5
$f(x)$	3	7	2	9	7

Thought Process

Goal: Determine if the function is one-to-one.

Strategy: Check if any output value appears more than once in the table.

Step-by-step:

Look at the output row: 3, 7, 2, 9, 7
Scan for repeats: I see 7 appears at $x = 2$ and $x = 5$
Two different inputs (2 and 5) give the same output (7)
This violates the one-to-one definition

Show Answer

No, the function is not one-to-one.

The output 7 appears twice: $f(2) = 7$ and $f(5) = 7$.

Since two different inputs (2 and 5) give the same output (7), the function fails the one-to-one test.

Level 2 Applying the Horizontal Line Test

Determine whether $g(x) = x^4 - 1$ is one-to-one.

Thought Process

Goal: Determine if $g(x) = x^4 - 1$ is one-to-one.

Two approaches:

Approach 1 (Algebraic): Find two different inputs with the same output.

Try $x = 1$: $g(1) = 1 - 1 = 0$
Try $x = -1$: $g(-1) = 1 - 1 = 0$
Same output, different inputs!

Approach 2 (Graphical): The graph of $x^4$ is U-shaped like $x^2$, so horizontal lines cross it twice.

Show Answer

No, $g(x) = x^4 - 1$ is not one-to-one.

For example:

$g(1) = 1^4 - 1 = 0$
$g(-1) = (-1)^4 - 1 = 0$

Since $g(1) = g(-1)$ but $1 \neq -1$, the function is not one-to-one.

Graphically: the graph of $x^4 - 1$ is U-shaped (like $x^2$), so horizontal lines cross it twice.

Level 3 Algebraic Proof

Prove algebraically that $f(x) = \frac{x}{x+2}$ (for $x \neq -2$) is one-to-one.

Thought Process

Assume $f(x_1) = f(x_2)$ and show this implies $x_1 = x_2$. Cross-multiply and simplify.

Show Answer

Assume $f(x_1) = f(x_2)$:

$$\frac{x_1}{x_1 + 2} = \frac{x_2}{x_2 + 2}$$

Cross-multiply:

$$x_1(x_2 + 2) = x_2(x_1 + 2)$$

$$x_1 x_2 + 2x_1 = x_2 x_1 + 2x_2$$

$$2x_1 = 2x_2$$

$$x_1 = x_2$$

Therefore $f$ is one-to-one. ✓

Level 4 Using Calculus

Show that $f(x) = x^3 + 2x + 1$ is one-to-one by analyzing its derivative.

Thought Process

If $f'(x) > 0$ for all $x$, then $f$ is strictly increasing, hence one-to-one. Compute $f'(x)$ and show it's always positive.

Show Answer

Compute the derivative:

$$f'(x) = 3x^2 + 2$$

Since $x^2 \geq 0$ for all $x$:

$$f'(x) = 3x^2 + 2 \geq 0 + 2 = 2 > 0$$

Since $f'(x) > 0$ for all $x$, the function $f$ is strictly increasing on $\mathbb{R}$.

A strictly increasing function is one-to-one. ✓

Level 5 Domain Restriction

The function $f(x) = x^2 - 4x + 3$ is not one-to-one on $\mathbb{R}$.

(a) Find the largest interval containing $x = 5$ on which $f$ is one-to-one.

(b) Find the largest interval containing $x = 0$ on which $f$ is one-to-one.

Thought Process

Goal: Find the largest one-to-one intervals containing $x = 5$ and $x = 0$.

Key insight: A parabola is one-to-one on each side of its vertex.

Step 1: Find the vertex of $f(x) = x^2 - 4x + 3$.

Complete the square: $f(x) = (x-2)^2 - 1$
Vertex is at $x = 2$

Step 2: Identify monotonic regions.

For $x \leq 2$: function is decreasing (one-to-one)
For $x \geq 2$: function is increasing (one-to-one)

Step 3: Match to the requested points.

$x = 5$ is in the increasing region ($x \geq 2$)
$x = 0$ is in the decreasing region ($x \leq 2$)

Show Answer

First, find the vertex. Complete the square or use $x = -b/(2a)$:

$$f(x) = x^2 - 4x + 3 = (x-2)^2 - 1$$

The vertex is at $x = 2$.

For $x \geq 2$: $f$ is increasing, hence one-to-one
For $x \leq 2$: $f$ is decreasing, hence one-to-one

(a) Since $5 > 2$, the largest interval containing $x = 5$ on which $f$ is one-to-one is $\boxed{[2, \infty)}$.

(b) Since $0 < 2$, the largest interval containing $x = 0$ on which $f$ is one-to-one is $\boxed{(-\infty, 2]}$.

CCI-Style Conceptual Questions

Question 1: A function $f$ has the property that $f(3) = 7$. Which of the following is possible if $f$ is one-to-one?

(A) $f(5) = 7$ (B) $f(-3) = 7$ (C) $f(3) = -7$ (D) $f(7) = 3$

Answer

(D) If $f$ is one-to-one, no other input can produce output 7. Options (A) and (B) would mean two inputs give output 7, violating one-to-one. Option (C) contradicts $f(3) = 7$. Option (D) is fine—nothing prevents $f(7) = 3$.

Question 2: If $f'(x) < 0$ for all $x$ in the domain of $f$, then $f$ is:

(A) One-to-one and increasing (B) One-to-one and decreasing (C) Not one-to-one (D) Cannot determine

Answer

(B) A function with $f'(x) < 0$ everywhere is strictly decreasing. Strictly decreasing functions are one-to-one (different inputs give different outputs because the function always goes down).

Mastery Checklist

[ ] I can state the definition of a one-to-one function
[ ] I can apply the horizontal line test to a graph
[ ] I can prove a function is one-to-one algebraically
[ ] I can identify whether common functions (polynomials, exponentials) are one-to-one
[ ] I can use derivatives to determine if a function is one-to-one
[ ] I can restrict domains to make functions one-to-one

Mental Model

The Unique Address Principle:

Think of a one-to-one function like a perfect mailing system: every output has exactly one "return address" (input). If two different people could send mail that arrives with the same label, you couldn't tell who sent it—that's not one-to-one.

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Last updated: 2026-01-22