One-to-one

Definition

A one-to-one function is a function in which each element of the range is paired with exactly one element of the domain. This implies that no two different inputs produce the same output, ensuring the function passes the horizontal line test.

5 Must Know Facts For Your Next Test

1. A one-to-one function has an inverse that is also a function.
2. Graphically, a function is one-to-one if any horizontal line intersects its graph at most once.
3. For exponential functions $f(x) = a^x$ where $a > 0$ and $a \neq 1$, they are always one-to-one.
4. Logarithmic functions $f(x) = \log_a(x)$ where $a > 0$ and $a \neq 1$ are also one-to-one.
5. The composition of two one-to-one functions is also a one-to-one function.

Review Questions

• What test can you use to determine if a function is one-to-one?
• Is the exponential function $f(x) = e^x$ one-to-one? If so, why?
• Can a logarithmic function have an inverse? Explain your answer.

Related terms

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

$\log_a(x)$: The logarithm of \( x \) with base \( a \), denoted as \( \log_a(x) \), is the exponent to which \( a \) must be raised to yield \( x \).

$e^x$: $e^x$ represents the exponential function with base \( e \), where \( e \approx 2.71828 \). It is commonly used in natural logarithms and growth processes.