We learned that a function is a rule that assigns one and only one value to each element of its domain. However a function may assign the same value to two or more elements in its domain. For example $f(x)=x^{2}$ assigns 4 to both $x=2$ and $x=-2$. Or $f(x)=|x|$ assigns the same value to both $x=a$ and $x=-a$. Or $f(x)=c$ where $c$ is a constant, takes on the same value $c$ at all $x$. But some functions assign distinct values to distinct elements of their domains. For example, $f(x)=2x+3$ takes on a different value at each value of $x$. Such functions are called one-to-one or 1-1 functions.

**Definition:**A function $f:A\to B$ is one-to-one (or an injection) if for all

$x_{1},x_{2}$ in $A$

\[

f(x_{1})=f(x_{2})\qquad\text{implies}\qquad x_{1}=x_{2}.\qquad{\small (\text{a})}

\]

- Equivalently we can say that a function is one-to-one whenever $x_{1}\neq x_{2}$ in $A$, then $f(x_{1})\neq f(x_{2})$.

\[

x_{1}\neq x_{2}\Rightarrow f(x_{1})\neq f(x_{2}).\qquad\quad{\small (\text{b})}

\] - The above definition states that a one-to-one function $y=f(x)$ takes on each value in its range only once. If the graph of the function is cut by a horizontal line $y=c$ at more than one point, the value

$y=c$ will correspond to more than one value of $x$, and the function will not be one-to-one. We may use the following test to specify whether or not a function is one-to-one.

**Horizontal line test for one-to-one functions:**

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.

Note that in the above example, although $f$ is not a one-to-one function on its entire natural domain $\mathbb{R}$, if we restrict the domain to $x\geq0$ , i.e. $f:[0,\infty)\to\mathbb{R}$ with $f(x)=x^{2}+1$, then the horizontal line test is passed and the function becomes one-to-one. Remark that the original and restricted functions are not the same functions because their domains are different. However, the two functions assume the same values on $[0,\infty)$.

- The function $f:(-\infty,0]\to\mathbb{R},$ $f(x)=x^{2}+1$ is also one-to-one.

We can figure out whether or not a function is one-to-one just by looking at their graphs. In the following table, we have investigated some common functions:

function | natural domain | 1-1 on natural domain? | Graphs |

$f(x)=x^n$ ($n$ is even) |
$\mathbb{R}$ | no | |

$f(x)=x^n$ ($n$ is odd) |
$\mathbb{R}$ | yes | |

$f(x)=\sqrt[n]{x}$ ($n$ is even) |
$[0,\infty)$ | yes | |

$f(x)=\sqrt[n]{x}$ ($n$ is odd) |
$\mathbb{R}$ | yes | |

$f(x)=\dfrac{1}{x^n}$ ($n$ is even) |
$\mathbb{R}-{0}=\{x|\ x\neq 0\}$ | no | |

$f(x)=\dfrac{1}{x^n}$ ($n$ is odd) |
$\mathbb{R}-{0}=\{x|\ x\neq 0\}$ | yes |

- Note that every
**monotonic function is one-to-one**. Recall that a monotonic function is a function that is increasing or decreasing. Suppose $f$ is an increasing function because if $x_{1}<x_{2}$ then $f(x_{1})<f(x_{2})$ and if $x_{2}<x_{1}$ then $f(x_{2})<f(x_{1})$. That is if $x_{1}\ne x_{2}$ then $f(x_{1})\neq f(x_{2})$. Similarly, we can show that if $f$ is a decreasing function, then it is one-to-one.