Editing Injective function (section)

== Examples ==
''For visual examples, readers are directed to the [[#Gallery|gallery section.]]''
* For any set <math>X</math> and any subset <math>S \subseteq X,</math> the [[inclusion map]] <math>S \to X</math> (which sends any element <math>s \in S</math> to itself) is injective. In particular, the [[identity function]] <math>X \to X</math> is always injective (and in fact bijective).
* If the domain of a function is  the [[empty set]], then the function is the [[empty function]], which is injective.
* If the domain of a function has one element (that is, it is a [[singleton set]]), then the function is always injective.
* The function <math>f : \R \to \R</math> defined by <math>f(x) = 2 x + 1</math> is injective.
* The function <math>g : \R \to \R</math> defined by <math>g(x) = x^2</math> is {{em|not}} injective, because (for example) <math>g(1) = 1 = g(-1).</math> However, if <math>g</math> is redefined so that its domain is the non-negative real numbers <nowiki>[0,+∞)</nowiki>, then <math>g</math> is injective.
* The [[exponential function]] <math>\exp : \R \to \R</math> defined by <math>\exp(x) = e^x</math> is injective (but not [[Surjective function|surjective]], as no real value maps to a negative number).
* The [[natural logarithm]] function <math>\ln : (0, \infty) \to \R</math> defined by <math>x \mapsto \ln x</math> is injective.
* The function <math>g : \R \to \R</math> defined by <math>g(x) = x^n - x</math> is not injective, since, for example, <math>g(0) = g(1) = 0.</math>

More generally, when <math>X</math> and <math>Y</math> are both the [[real line]] <math>\R,</math> then an injective function <math>f : \R \to \R</math> is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the {{em|[[horizontal line test]]}}.<ref name=":0" />