One-to-One Functions

Test	Description
Algebraic	$f(x_1)=f(x_2)\Rightarrow x_1=x_2$
Graphical	Every horizontal line meets the graph at most once

Definition

We learned that a function assigns one and only one output value to each element of its domain. However, a function may assign the same output to two or more different inputs. For example:

• $f(x)=x^2$ assigns 4 to both $x=2$ and $x=-2$.
• $f(x)=|x|$ assigns the same value to $x=a$ and $x=-a$.
• $f(x)=c$ (a constant) assigns $c$ to every $x$.

Some functions, however, assign distinct outputs to distinct inputs. For example, $f(x)=2x+3$ takes a different value at each value of $x$. Such functions are called one-to-one.

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

The 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 $x$-value, and the function is not one-to-one. This gives us the visual test:

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

• A linear function $f(x)=mx+b$ (with $m\ne 0$) is one-to-one because every horizontal line meets it exactly once.
• The function $f(x)=x^2+1$ is not one-to-one on $ℝ$ because a horizontal line $y=c$ (for ) meets its graph at two points.

Frequently Asked Questions

Can a function that is not one-to-one on all of $ℝ$ be made one-to-one by restricting its domain?

Yes. For example, $f(x)=x^2$ is not one-to-one on $ℝ$, but it is one-to-one on $[0,+\mathrm{\infty })$. Restricting the domain to $[0,+\mathrm{\infty })$ makes it one-to-one and therefore invertible (the inverse is $f^{-1}(x)=\sqrt{x}$). This strategy is exactly how inverse trigonometric functions are defined: we restrict the domain of the trig function to make it one-to-one.

Are all increasing functions one-to-one?

Yes. If $f$ is increasing on an interval $I$, then implies , which means $f(x_1)\ne f(x_2)$ whenever $x_1\ne x_2$. The same argument works for decreasing functions. So every strictly monotone function is one-to-one.

What is the connection between one-to-one functions and having an inverse?

A function has an inverse if and only if it is one-to-one. The reason: to define $f^{-1}(y)=x$, we need to know that there is a unique $x$ satisfying $f(x)=y$. If $f$ is not one-to-one, there could be two different $x$-values mapping to the same $y$, and $f^{-1}(y)$ would be ambiguous.