One-to-One Functions

A function is one-to-one if it never assigns the same output to two different inputs. This property is essential for defining inverse functions and is easily detected from a graph using the horizontal line test.

Quick Reference

Concept	Description
One-to-one (injective)	$f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$
Equivalent condition	$x_{1} \neq x_{2} \Rightarrow f (x_{1}) \neq f (x_{2})$
Horizontal line test	Every horizontal line meets the graph at most once
Monotonic $\Rightarrow$ one-to-one	Every increasing or decreasing function is injective

What Is a One-to-One Function?

A function may assign the same output value to more than one input. For example, $f (x) = x^{2}$ assigns the value $4$ to both $x = 2$ and $x = - 2$ . Similarly, $f (x) = | x |$ gives the same output for $x = a$ and $x = - a$ , and a constant function $f (x) = c$ takes the same value everywhere.

By contrast, $f (x) = 2 x + 3$ produces a distinct output for every input. Such functions are called one-to-one (or injective) functions.

A function $f : A \to B$ is one-to-one (or an injection) if for all $x_{1}, x_{2}$ in $A$ :

Equivalently, whenever $x_{1} \neq x_{2}$ in $A$ :

In other words, a one-to-one function takes on each value in its range exactly once.

Horizontal Line Test

If the graph of $y = f (x)$ is cut by a horizontal line $y = c$ at more than one point, then the value $c$ corresponds to more than one input, and the function is not one-to-one. This gives a quick visual test:

Horizontal Line Test: A function is one-to-one if and only if every horizontal line $y = c$ intersects the graph of $y = f (x)$ at most once.

Examples

Let $f (x) = m x + b$ ( $m \neq 0$ ). Is $f$ a one-to-one function?

Solution

Method 1 (algebraic): If

m x_{1} + b = m x_{2} + b

, then

m x_{1} = m x_{2}

, so

x_{1} = x_{2}

. Thus

f

is one-to-one. Method 2 (graphical): Each horizontal line

y = c

intersects the graph of a non-horizontal line exactly once, so

f

passes the horizontal line test.

Let $f (x) = x^{2} + 1$ . Is $f$ a one-to-one function?

Solution

Method 1 (algebraic): If

x_{1}^{2} + 1 = x_{2}^{2} + 1

, then

x_{1}^{2} = x_{2}^{2}

, so

x_{1} = x_{2}

x_{1} = - x_{2}

. Because two different inputs can give the same output,

f

is not one-to-one. Method 2 (graphical): A horizontal line

y = c

with

c > 1

intersects the parabola at two points, failing the horizontal line test.

Graph of the parabola y = x² + 1 illustrating it fails the horizontal line test (not one-to-one). — $y = x^{2} + 1$ is not one-to-one on its entire domain $ℝ$ .

Restricting the domain: Although

f

is not one-to-one on all of

ℝ

, if we restrict to

x \geq 0

, the function

f : [0, \infty) \to ℝ

with

f (x) = x^{2} + 1

passes the horizontal line test and becomes one-to-one. (Similarly, the restriction to

x \leq 0

is also one-to-one.)

Graph of y = x² + 1 restricted to x ≥ 0, showing it passes the horizontal line test. — Restricting the domain of $f (x) = x^{2} + 1$ to $[0, \infty)$ makes it one-to-one: each horizontal line meets the graph at most once.

Let $f : ℝ - {0} \to ℝ$ , where $f (x) = 1 / x$ . Is $f$ a one-to-one function?

Solution

Method 1 (algebraic): If

1 / x_{1} = 1 / x_{2}

, then

x_{1} = x_{2}

. So

f

is one-to-one. Method 2 (graphical): Each horizontal line

y = c

with

c \neq 0

intersects the hyperbola exactly once, and

y = 0

never intersects it.

Graph of the hyperbola y = 1/x showing it passes the horizontal line test. — Each horizontal line intersects the graph of $y = 1 / x$ at most once.

Summary Table of Common Functions

Function	Natural Domain	One-to-one?
$f (x) = x^{n}$ ( $n$ even)	$ℝ$	No
$f (x) = x^{n}$ ( $n$ odd)	$ℝ$	Yes
$f (x) = \sqrt[n]{x}$ ( $n$ even)	$[0, \infty)$	Yes
$f (x) = \sqrt[n]{x}$ ( $n$ odd)	$ℝ$	Yes
$f (x) = 1 / x^{n}$ ( $n$ even)	$ℝ - {0}$	No
$f (x) = 1 / x^{n}$ ( $n$ odd)	$ℝ - {0}$	Yes

Every monotonic function is one-to-one. If $f$ is increasing and $x_{1} \neq x_{2}$ , then either $x_{1} < x_{2}$ (so $f (x_{1}) < f (x_{2})$ ) or $x_{1} > x_{2}$ (so $f (x_{1}) > f (x_{2})$ ); either way, $f (x_{1}) \neq f (x_{2})$ . The same argument applies to decreasing functions.

Frequently Asked Questions

What does "one-to-one" mean in plain language?

A one-to-one function never sends two different inputs to the same output. Think of it as a strict pairing: each output value is matched to exactly one input value.

How is the horizontal line test different from the vertical line test?

The vertical line test checks whether a curve represents a function at all (each input has at most one output). The horizontal line test checks whether a function is one-to-one (each output comes from at most one input). You apply the vertical line test first; only after that does it make sense to apply the horizontal line test.

Can I make any function one-to-one by restricting its domain?

Yes, in principle. For example,

f (x) = x^{2}

is not one-to-one on

ℝ

, but it becomes one-to-one on

[0, \infty)

or on

(- \infty, 0]

. However, the restricted function is a different function from the original, because its domain is different.

Why does one-to-one matter for inverse functions?

A function has an inverse if and only if it is one-to-one. If two different inputs

x_{1} \neq x_{2}

give the same output

y

, then we cannot define

f^{- 1} (y)

unambiguously: it would need to equal both

x_{1}

and

x_{2}

, which is not allowed for a function.

Quick Reference

What Is a One-to-One Function?

Horizontal Line Test

Examples

Summary Table of Common Functions

Frequently Asked Questions

Quick Links

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