Injective Function - Examples

Examples

• For any set X and any subset S of X the inclusion map S → X (which sends any element s of S to itself) is injective. In particular the identity function X → X is always injective (and in fact bijective).
• If the domain X = ∅ or X has only one element, the function X → Y is always injective.
• The function f : R → R defined by f(x) = 2x + 1 is injective.
• The function g : R → R defined by g(x) = x2 is not injective, because (for example) g(1) = 1 = g(−1). However, if g is redefined so that its domain is the non-negative real numbers [0,+∞), then g is injective.
• The exponential function exp : R → R defined by exp(x) = ex is injective (but not surjective as no real value maps to a negative number).
• The natural logarithm function ln : (0, ∞) → R defined by x ↦ ln x is injective.
• The function g : R → R defined by g(x) = xn − x is not injective, since, for example, g(0) = g(1).

More generally, when X and Y are both the real line R, then an injective function f : R → R is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the horizontal line test.