One-to-One Functions

Increasing and Decreasing FunctionsInverse 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)
Equivalent condition
Horizontal line test Every horizontal line meets the graph at most once
Monotonic 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, assigns the value to both and . Similarly, gives the same output for and , and a constant function takes the same value everywhere.

By contrast, produces a distinct output for every input. Such functions are called one-to-one (or injective) functions.

A function is one-to-one (or an injection) if for all in :

Equivalently, whenever in :

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

Horizontal Line Test

If the graph of is cut by a horizontal line at more than one point, then the value 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 intersects the graph of at most once.

Examples

Let (). Is a one-to-one function?

Solution Method 1 (algebraic): If , then , so . Thus is one-to-one. Method 2 (graphical): Each horizontal line intersects the graph of a non-horizontal line exactly once, so passes the horizontal line test.
Graph of a linear function y = mx + b illustrating it passes the horizontal line test (one-to-one).
Each horizontal line intersects the graph of a linear function exactly once.

Let . Is a one-to-one function?

Solution Method 1 (algebraic): If , then , so or . Because two different inputs can give the same output, is not one-to-one. Method 2 (graphical): A horizontal line with 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).
is not one-to-one on its entire domain .
Restricting the domain: Although is not one-to-one on all of , if we restrict to , the function with passes the horizontal line test and becomes one-to-one. (Similarly, the restriction to 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 to makes it one-to-one: each horizontal line meets the graph at most once.

Let , where . Is a one-to-one function?

Solution Method 1 (algebraic): If , then . So is one-to-one. Method 2 (graphical): Each horizontal line with intersects the hyperbola exactly once, and never intersects it.
Graph of the hyperbola y = 1/x showing it passes the horizontal line test.
Each horizontal line intersects the graph of at most once.

Summary Table of Common Functions

Function Natural Domain One-to-one?
( even) No
( odd) Yes
( even) Yes
( odd) Yes
( even) No
( odd) Yes

Every monotonic function is one-to-one. If is increasing and , then either (so ) or (so ); either way, . 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, is not one-to-one on , but it becomes one-to-one on or on . 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 give the same output , then we cannot define unambiguously: it would need to equal both and , which is not allowed for a function.
Increasing and Decreasing FunctionsInverse Functions
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