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The function $x^{2}$ with domain $\mathbb {R}$ does not have an inverse function. If we restrict $x^{2}$ to the non-negative real numbers, then it does have an inverse function, known as the square root of $x.$

Function
$x \mapsto f (x)$
History of the function concept
Types by domain and codomain
$X \to 𝔹$ $𝔹 \to X$ $𝔹 n \to X$ $X \to ℤ$ $ℤ \to X$ $X \to ℝ$ $ℝ \to X$ $ℝ n \to X$ $X \to ℂ$ $ℂ \to X$ $ℂ n \to X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
List of specific functions
v t e

In mathematics, the restriction of a function $f$ is a new function, denoted $f\vert _{A}$ or $f{\upharpoonright _{A}},$ obtained by choosing a smaller domain $A$ for the original function $f.$ The function $f$ is then said to extend $f\vert _{A}.$