One-to-one function

Definition

A one-to-one function is a function where each input corresponds to exactly one unique output, and each output corresponds to exactly one unique input. In other words, no two different inputs produce the same output.

5 Must Know Facts For Your Next Test

1. A function $f(x)$ is one-to-one if and only if $f(a) \neq f(b)$ whenever $a \neq b$.
2. The horizontal line test can be used to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, it is not one-to-one.
3. One-to-one functions have inverses that are also functions.
4. If $f(x)$ is one-to-one, then it passes the vertical line test (since all functions do) and the horizontal line test.
5. In trigonometry, some trigonometric functions are restricted to specific intervals to become one-to-one so their inverses can be defined.

Review Questions

• How can you use the horizontal line test to determine if a function is one-to-one?
• What does it mean for a function to have an inverse in terms of being one-to-one?
• Why might certain trigonometric functions need to be restricted to specific intervals?

Related terms

Inverse Function: A function that reverses the operation of another function. If $f$ maps $x$ to $y$, then its inverse $f^{-1}$ maps $y$ back to $x$.

Horizontal Line Test: A method used to determine if a function is one-to-one. A function passes this test if every horizontal line intersects its graph at most once.

$\text{Arcsin}$: $\text{Arcsin}$ or inverse sine is the inverse of the sine function, typically defined on the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$ for it to be one-to-one.