One-to-One Function
Functions & Advanced AlgebraA one-to-one function is a function in which every output value corresponds to exactly one input value.
Definition
A one-to-one function is a special kind of function where no two different inputs give the same output. Every output is unique.
Example
$f(x) = 2x$ is one-to-one: different inputs always give different outputs ($f(3) = 6$, $f(4) = 8$, never the same). But $f(x) = x^2$ is NOT: $f(3) = 9$ AND $f(-3) = 9$.
Key Insight
One-to-one functions are the only functions that can be "undone" perfectly with an inverse function. If two inputs can produce the same output, you cannot reverse the process uniquely.
Definition
A function $f$ is one-to-one (injective) if $f(a) = f(b)$ implies $a = b$ for all $a, b$ in the domain. Graphically, it passes the horizontal line test: every horizontal line crosses the graph at most once.
Example
$f(x) = 3x - 5$ is one-to-one (proof: $f(a) = f(b)$ means $3a - 5 = 3b - 5$, so $a = b$). $f(x) = x^2$ is not on all reals, but is one-to-one if restricted to $x \ge 0$.
Key Insight
One-to-one functions are precisely the functions that have inverses. Restricting the domain of a non-one-to-one function can make it one-to-one (standard practice with $f(x) = x^2$).
Definition
A function $f: A \to B$ is injective if for all $a_1, a_2 \in A$, $f(a_1) = f(a_2)$ implies $a_1 = a_2$. Equivalently, distinct elements of $A$ map to distinct elements of $B$. An injective function from $A$ to $B$ implies $|A| \le |B|$ for finite sets.
Example
The function $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(n) = 2n$ is injective (even integers only form the image, so it is not surjective onto $\mathbb{Z}$). The composition of two injections is injective.
Key Insight
Injectivity, surjectivity, and bijectivity form the foundational classification of functions in set theory. A bijection between two sets proves they have the same cardinality, even for infinite sets.