Composition of functions

Let f : X → Y1 and g : Y₂ → Z be two functions and the set D = {x ∈ X : f (x) ∈ Y2). Then the function h defined on D by h(x) = g(f(x)) is called composite function of g and f and is denoted by gοf. It is also called function of a function.

Remarks:

(i) Domain of gof is D which is a subset of Y2 (the domain of g ).

(ii) Range of gof is a subset of range of g.

(iii) If D = X, then f(x) ⊂ Y2.

Note that gof is defined only if ∀ x ∈ X, f(x) is an element of the domain of g so that we can take its g-image. Hence for the product gof of two functions f & g, the range of f must be a subset of

the domain of g.

Some Results

(1) If both f and g are one-one then so is gof.

(2) Let f : X → Y and g : Y → Z. Then

(i) If both f and g are onto then gof is onto.

(ii) If gof is one-one, then f is one-one. But g may not be one-one.

(iii) If gof is onto, then g is onto. But f may not be onto.