One-to-One Functions

A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain.

Learning Objective

Identify the properties of a one-to-one function

Key Points

A one-to-one function has a unique output for each unique input.
Domain restriction can allow a function to become one-to-one, such as in the case of $f(x)=x^2$ for $x\geq 0$.
To check if a function is a one-to-one perform the horizontal line test. If any horizontal line intersects the graph in more than one point, the function is not one-to-one.
If every element of a function's range corresponds to exactly one element of its domain, then the function is said to be one-to-one.

Term

injective function
A function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

Full Text

Properties of a One-To-One Function

A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its co-domain. In other words, every element of the function's range corresponds to exactly one element of its domain. Occasionally, an injective function from $X$ to $Y$ is denoted $f: X \mapsto Y$, using an arrow with a barbed tail.

An easy way to check if a function is a one-to-one is by graphing it and then performing the horizontal line test. If any horizontal line intersects the graph at more than one point, the function is not one-to-one. To see this, note that the points of intersection have the same y-value, because they lie on the line, but different x values, which by definition means the function cannot be one-to-one.

Horizontal Line Test

Because the horizontal line crosses the graph of the function more than once, it fails the horizontal line test and cannot be one-to-one.

Horizontal line test fail.

Determining If a Function is One-To-One

Example 1: Is the function $f(x)={x}^{2}$ (with no domain restrictions) one-to-one?

One way to check if the function is one-to-one is to graph the function and perform the horizontal line test. The graph below shows that it forms a parabola and fails the horizontal line test.

Parabola Graph

The graph of the function $f(x)=x^2$ fails the horizontal line test and is therefore NOT a one-to-one function. If a horizontal line can go through two or more points on the function's graph then the function is NOT one-to-one.

Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain. A list of ordered pairs for the function are:

$\displaystyle (-2,4)\\ (-1,1)\\ (0,0)\\ (1,1)\\ (2,4)$

The ordered pairs $(-2,4)$ and $(2,4)$ do not pass the definition of one-to-one because the element $4$ of the range corresponds to to $-2$ and $2$. Each unique input must have a unique output so the function cannot be one-to-one. Notice also, that these two ordered pairs form a horizontal line; which also means that the function is not one-to-one as stated earlier.

Example 2: Is the function $f(x)=\left | x-2 \right |$ one-to-one?

This is an absolute value function, which is graphed below. Notice it fails the horizontal line test. Because each unique input does not have a unique output, this function cannot be one-to-one.

Absolute Value Graph

The graph of the function,$f(x)=\left | x-2 \right |$, fails the horizontal line test and is therefore not a one-to-one function.

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