One-to-One Functions: Definition, Horizontal Line Test & Examples

Q: What is a one-to-one function?

A function f is one-to-one (injective) if every distinct input produces a distinct output. Equivalently, f(x₁) = f(x₂) implies x₁ = x₂ for all x₁ and x₂ in the domain.

Q: Are all linear functions one-to-one?

Every non-constant linear function f(x) = ax + b with a ≠ 0 is one-to-one. A constant function (a = 0) is not.

Q: Is f(x) = x² a one-to-one function?

No. Because f(-1) = f(1) = 1, two distinct inputs give the same output. Restricting the domain to x ≥ 0 would make it one-to-one.

Q: Why do only one-to-one functions have inverse functions?

If two distinct inputs x₁ and x₂ mapped to the same output y, then an inverse would need to map y back to both x₁ and x₂ — which violates the definition of a function. A one-to-one function avoids this, so its inverse is also a function.

Q: How do you prove algebraically that a function is one-to-one?

Assume f(x₁) = f(x₂) and use algebra to deduce x₁ = x₂. This is the contrapositive of the definition and is usually the most convenient form for a proof.

Understanding Functions and Injectivity

A function is a rule that assigns to each element of a set called the domain exactly one element of another set called the codomain; the set of values actually produced is the range.

A function is one-to-one (injective) if no two distinct elements of the domain map to the same element of the range. One-to-one functions are central to the study of inverse functions and to solving certain classes of equations.

Algebraic definition

Let \( x_1 \) and \( x_2 \) be any elements of the domain. A function \( f \) is one-to-one if:

\[ x_1 \ne x_2 \;\Rightarrow\; f(x_1) \ne f(x_2) \]

Equivalently, using the contrapositive — usually far more useful for proofs:

\[ f(x_1) = f(x_2) \;\Rightarrow\; x_1 = x_2 \]

Visualizing One-to-One Functions

The Venn diagram below shows a one-to-one function: no two input values share the same output value.

Venn diagram of a one-to-one function mapping unique inputs to unique outputs — **Figure 1.** Venn diagram of a one-to-one function.

In contrast, the next diagram shows a function that is not one-to-one: the inputs \(-1\) and \(0\) both map to the same output.

Venn diagram of a function that is not one-to-one where multiple inputs map to a single output — **Figure 2.** Venn diagram of a function that is not one-to-one.

The Horizontal Line Test

When the graph is available, the horizontal line test gives a quick check: the function is one-to-one if and only if every horizontal line intersects the graph at most once.

The following graph is not one-to-one: a horizontal line can intersect the curve in more than one place, so distinct inputs share the same output.

Graph of a function failing the horizontal line test — **Figure 4.** Graph of a function that is not one-to-one.

Examples of One-to-One Functions

Example 1 — Every linear function with nonzero slope is one-to-one

Show algebraically that every linear function \( f(x) = ax + b \) with \( a \ne 0 \) is one-to-one.

Solution:

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

\[ ax_1 + b = ax_2 + b \] \[ a(x_1 - x_2) = 0 \]

Since the slope \( a \ne 0 \), dividing by \( a \) gives:

\[ x_1 = x_2 \]

Therefore every non-constant linear function is one-to-one.

Example 2 — A quadratic is not one-to-one on its full domain

Show analytically and graphically that \( f(x) = -x^2 + 3 \) is not one-to-one.

Solution:

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

\[ -x_1^2 + 3 = -x_2^2 + 3 \] \[ -(x_1^2 - x_2^2) = 0 \] \[ -(x_1 - x_2)(x_1 + x_2) = 0 \]

This gives two possibilities:

\[ x_1 = x_2 \quad \text{or} \quad x_1 = -x_2 \]

Equality of outputs does not force \( x_1 = x_2 \) — for example, \( f(2) = f(-2) = -1 \). Hence the function is not one-to-one.

Graph of y = -x² + 3 failing the horizontal line test — **Figure 5.** Quadratic function failing the horizontal line test.

Example 3 — Common one-to-one functions

Show graphically that \( f(x) = \ln(x) \), \( g(x) = e^x \), and \( h(x) = x^3 \) are one-to-one.

Each graph below intersects any horizontal line at most once, so each function passes the horizontal line test.

Graph of the natural logarithm ln(x) passing the horizontal line test — **Figure 6.** \( f(x) = \ln(x) \).

Graph of the exponential function e^x passing the horizontal line test — **Figure 7.** \( g(x) = e^x \).

Graph of the cubic function x³ passing the horizontal line test — **Figure 8.** \( h(x) = x^3 \).

How One-to-One Functions Are Used

1. Finding inverse functions

A function has an inverse function if and only if it is one-to-one. The inverse of a one-to-one function is itself one-to-one.

2. Solving equations

Because logarithmic and exponential functions are one-to-one, if their outputs are equal, their inputs must be equal — provided both sides are in the domain.

\[ \ln(2x + 3) = \ln(4x - 2) \;\Rightarrow\; 2x + 3 = 4x - 2 \quad (\text{requires } x > \tfrac{1}{2}) \] \[ e^{-x + 2} = e^{3x - 8} \;\Rightarrow\; -x + 2 = 3x - 8 \]

One-to-One at a Glance

Summary of function types and whether they are one-to-one
Function	One-to-One?	Why
\( ax + b \), \( a \ne 0 \)	Yes	Nonzero slope
Constant	No	All inputs give one output
\( x^2 \) on all reals	No	\( f(-1) = f(1) \)
\( x^2 \) on \( x \ge 0 \)	Yes	Strictly increasing
\( x^3 \), \( e^x \), \( \ln x \)	Yes	Strictly monotonic

Frequently Asked Questions

What is a one-to-one function?

A function \( f \) is one-to-one (injective) if every distinct input produces a distinct output. Equivalently, \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \) for all \( x_1, x_2 \) in the domain.

What is the horizontal line test?

A function is one-to-one if and only if every horizontal line intersects its graph at most once. If any horizontal line crosses the graph two or more times, the function is not one-to-one.

Are all linear functions one-to-one?

Every non-constant linear function \( f(x) = ax + b \) with \( a \neq 0 \) is one-to-one. A constant function (\( a = 0 \)) is not.

Is f(x) = x² a one-to-one function?

No. Because \( f(-1) = f(1) = 1 \), two distinct inputs give the same output. Restricting the domain to \( x \ge 0 \) would make it one-to-one.

Why do only one-to-one functions have inverse functions?

If two distinct inputs \( x_1 \) and \( x_2 \) mapped to the same output \( y \), an inverse would have to map \( y \) back to both \( x_1 \) and \( x_2 \) — which violates the definition of a function. One-to-one functions avoid this, so their inverse is also a function.

How do you prove algebraically that a function is one-to-one?

Assume \( f(x_1) = f(x_2) \) and use algebra to deduce \( x_1 = x_2 \). This contrapositive form is usually the most convenient for a proof.