# 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.

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:

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.

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