### Functions

### Lesson 4: Composite Function

(1) The meaning of "composite?

A composite function is made up of several functions in a sequential order. For example gf represents a composite function whereby function f is applied first followed by function g.

Example 1

(2) Existence of composite function

A composite function exists if the range of the first function is a subset of the domain of the second function.

E.g. The composite function gf exists if .

(3) Domain & Rule of composite function gf

(4) Range of composite function gf

Method 1:

Sketch the graph of gf (x) under the domain of gf (= domain of f) and obtained the range from the set of y values from the graph.

Method 2 (mapping method):

Sketch the graph of the second function, g under the domain given by the range of the first function. Then the range of the composite is the set of y values from the graph.

Example 2

Given two functions f and g defined as follows:

Show that the composite gf exists and find its domain, rule and range.

Solution