Composition of Functions

Let ƒ: A➞B and g: B➞C be any two functions. Suppose x∈A, then its image ƒ(x)∈B. B being the domain of g, we can find the image of ƒ(x)∈B under g, i.e., g(ƒ(x))∈C. In other words, we can associate an element x∈A with a unique element g(ƒ(x))∈C. Hence, we have a function from A to C. This new function is known as the composite function of ƒ and g (not g and ƒ). It is denoted by goƒ or g(ƒ).

If ƒ: A➞B and g: B➞C be any two functions, then the composite function of ƒ and g is the function goƒ: A➞C and defined by the equation, \[(goƒ)(x)=g(ƒ(x))\] Composite function is also known as product function or function of a function.

Composite Function of f and g

Examples:

  1. Let A, B and C denote the sets of real numbers. Suppose ƒ: A➞B and g: B➞C are defined by \[ƒ(x)=x-1\] \[g(x)=x^2\] Then, \[(goƒ)(x)=g(ƒ(x))\] \[=g(x-1)\] \[=(x-1)^2\]
  2. Suppose ƒ: A➞B and g: B➞C are defined by the following diagram:
Example of composite function of f and g

Here, \[(goƒ)(a)=g(ƒ(a))=g(y)=q\] \[(goƒ)(b)=g(ƒ(b))=g(y)=q\] \[(goƒ)(c)=g(ƒ(c))=g(z)=p\]

Properties of Composite Functions

  1. If ƒ: A➞B, g: B➞C and h: C➞D, then \[(ho(goƒ))(x)=h((goƒ)(x))=h(g(ƒ(x)))\] \[((hog)oƒ)(x)=(hog)(ƒ(x))=h(g(ƒ(x)))\] \[\text{for any x∈A.}\] \[\therefore ho(goƒ)=(hog)oƒ\] Hence, composite function satisfies associative property.
  2. If ƒ: A➞B and g: B➞C are given functions, then goƒ is onto or one-one according as each of ƒ and g is onto or one-one.

If ƒ is onto, ƒ(A)=B, and so for any x∈A, ƒ(x)∈B. Since g is onto, g(B)=C for any ƒ(x)∈B, g(ƒ(x))∈C. But \[g(ƒ(x))=(goƒ)(x)\] \[\therefore (goƒ)(A)=C\] Hence, goƒ is onto.

If ƒ is one-one, every element of A has a distinct image in B. And, since g is one-one, every element of B has a distinct image in C. So, for any x∈A, goƒ(x) is unique. Hence, goƒ is one-one.

More on Relations And Functions

© 2022 AnkPlanet | All Rights Reserved