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If there are two functions:** f(x) **and **g(x)**, and the 2nd domain is within the range of the 1st, it can be defined as a new function that associates each element of the domain of **f(x)** with the value of **g[f(x)]**.

**(g o f) (x) = g [f(x)] = g (2x) = 3 (2x) +1 = 6x + 1**

(g o f) (1) = 6 · 1 + 1 = 7

## Domain

**D _{(g o f) } = {x D_{f } / f(x) D_{g}} **

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## Properties

Associative:

**f o (g o h) = (f o g) o h**

Not commutative.

**f o g ≠ g o f**

The identity element is the identity function, **i(x) = x**.

**f o i = i o f = f**

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