A one-to-one function (injective function) is a function where each element of the domain maps to a unique element in the codomain. No two different elements in the domain map to the same element in the codomain.
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A function $f$ is one-to-one if and only if $f(a) = f(b)$ implies $a = b$ for all elements $a$ and $b$ in its domain.
The horizontal line test can determine if a function is one-to-one: if every horizontal line intersects the graph at most once, then the function is one-to-one.
One-to-one functions have inverses that are also functions.
If a function is both one-to-one and onto, it is called bijective.
In composition of functions, if $g \circ f$ is one-to-one, then $f$ must be one-to-one.
Review Questions
What property must hold true for all pairs of elements in the domain of a one-to-one function?
How can you use the horizontal line test to determine if a function is one-to-one?
A function that reverses another function; if $f(x)$ maps $x$ to $y$, then its inverse $f^{-1}(y)$ maps $y$ back to $x$. Both must be functions.
Onto Function: $A \text{function} \ f: X \rightarrow Y \ \text{is onto (surjective)} \ \text{if every element} \ y \in Y \ \text{has a preimage} \ x\in X.$
Bijective Function: $A\text{function that is both injective (one-to-one) and surjective (onto).}$