Suppose f and g are two functions with respective domains X and Y, then g is called a extension of fto Y. if.

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BO X e Y and if f(x) g(x). Vxex

YCX and if f(x) g(x). VxEY

Xc Y and if f(x) = g(x). Vxe X

XcY and i

By Charlotte

Answer:In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z.

Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.[nb 1] The notation g ∘ f is read as “g circle f “, “g round f “, “g about f “, “g composed with f “, “g after f “, “g following f “, “g of f”, “f then g”, or “g on f “. Intuitively, composing functions is a chaining process in which the output of function f feeds the input of function g.

The composition of functions is a special case of the composition of relations, sometimes also denoted by {\displaystyle \circ }\circ .[1] As a result, all properties of composition of relations are true of composition of functions,[2] though the composition of functions has some additional properties.

Composition of functions is different from multiplication of functions, and has quite different properties;[3] in particular, composition of functions is not commutative.

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