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Composite Function

Let A, B, and C be three non-empty sets.
Let f: A B and g: B C be two functions then gof: A C. This function is called composition of f and g, given by gof (x) = g(f(x)) x ∈ A.
Thus the image of every x A under the function gof is the g-image of the f-image of x.
The gof is defined only if x A, f(x) is an element of the domain of g so that we can take its g-image.
The range of f must be a subset of the domain of g in gof.

Properties of composite functions

  1. The composition of function is not commutative, i.e., foggof.
  2. The composition of function is associative, i.e., if h: A B, g: B C and f: C D be three functions, then (fog)oh = fo(goh).
  3. The composition of any function with the identity function is the function itself, i.e., f: A B then foIA = IBof = f where IA and IB are the identity functions of A and B, respectively.

Identical Function

Two functions f and g are said to be identical if
  1. The domain of f = the domain of g, i.e., Df = Dg
  2. The range of f = the range of g
  3. f(x) = g(x) x Df or x Dg, e.g., f(x) = x and g(x) = 77922.png are not identical functions as Df = Dg, but Rf = R, Rg = [0, ∞)

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