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One-to-One Functions

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Reference: Stewart §6.1

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?

  1. What is a function? A rule that assigns exactly one output to each input.
  2. What does $f(3) = 7$ mean? When the input is 3, the output is 7.
  3. 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

PrerequisitesFunction DefinitionGraphing Functions
This skillOne-to-One Functions
Leads toExponential Functions and PropertiesFinding Inverse FunctionsInverse Function DefinitionLogarithmic Functions

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:

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

  1. Look at the output row: 3, 7, 2, 9, 7
  2. Scan for repeats: I see 7 appears at $x = 2$ and $x = 5$
  3. Two different inputs (2 and 5) give the same output (7)
  4. 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).

Common Misconceptions

Common misconception

a function is one-to-one if every output is positive or if the function is increasing on some interval.

This is the concept-image-conflicts-definition error. The one-to-one property requires that different inputs produce different outputs across the entire domain, not merely on part of it. The function $f(x) = x^2 - 4x$ is increasing on $[2, \infty)$, yet it is not one-to-one on all of $\mathbb{R}$ because $f(0) = 0 = f(4)$. The horizontal line test applied to the complete graph, not a restricted piece, is the reliable check.

Mastery Checklist

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