One-to-one function

Definition

A one-to-one function, also known as an injective function, is a type of function in which every element of the domain maps to a unique element in the codomain. This implies that no two different inputs can map to the same output.

5 Must Know Facts For Your Next Test

1. A function $f$ is one-to-one if and only if $f(a) \neq f(b)$ whenever $a \neq b$ for all $a$ and $b$ in its domain.
2. The horizontal line test can determine whether a function is one-to-one: if any horizontal line intersects the graph of the function at most once, then the function is one-to-one.
3. For any function to have an inverse that is also a function, it must be one-to-one.
4. The notation for the inverse of a one-to-one function $f$ is $f^{-1}$.
5. In calculus, many common functions like linear functions (with non-zero slopes) and exponential functions are typically one-to-one.

Review Questions

• How can you use the horizontal line test to verify if a function is one-to-one?
• Why must a function be one-to-one to have an inverse that is also a function?
• Give an example of a common mathematical function that is not one-to-one and explain why.

Related terms

Inverse Function: A function that reverses another function; if $f(x)$ maps $x$ to $y$, then its inverse $f^{-1}(y)$ maps $y$ back to $x$.

Horizontal Line Test: A method used to determine if a function is one-to-one by checking if any horizontal line intersects the graph of the function more than once.

Injective Function: (Another name for One-to-One Function) A type of mapping where each element of the domain maps to a unique element in the codomain.