One-to-one

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. This ensures that no two different inputs produce the same output.

5 Must Know Facts For Your Next Test

1. A one-to-one function passes the Horizontal Line Test; if any horizontal line intersects the graph of the function at most once, the function is one-to-one.
2. For exponential functions $f(x) = a^x$, where $a > 0$ and $a \neq 1$, they are always one-to-one.
3. The inverse of a one-to-one function is also a function. For example, if $f(x)$ is one-to-one, then there exists an inverse function $f^{-1}(x)$.
4. In logarithmic functions, such as $g(x) = \log_a(x)$ where $a>0$ and $a\neq 1$, they are inherently one-to-one.
5. To determine if a logarithmic or exponential equation represents a one-to-one function, you can check if it satisfies both injective (one-to-one) and surjective (onto) properties.

Review Questions

• What test can you use to determine if a function is one-to-one?
• Is the exponential function $f(x) = 2^x$ considered one-to-one? Why or why not?
• How does knowing that a function is one-to-one help in finding its inverse?

Related terms

Horizontal Line Test: A graphical method used to determine if 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.

Injective Function: A type of function where each element of the range is mapped by exactly one element of its domain, ensuring no two distinct inputs produce the same output.

Inverse Function: A function that reverses another function; for a given one-to-one function $f$, its inverse $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ for all x in the domain of $f^{-1}$.