02Icomposite

toc = **Composite Functions** =

A composite function is formed when one function is put “inside” another. Each x value of the outside function is substituted with the inside function.

** Example 1 **
If f(x) = sin(x) and g(x) = 2x + 1

then

... ... f( g(x) ) = sin( 2x + 1 )

OR

... ... g( f(x) ) = 2 sin(x) + 1

Notation
f(g(x)) is read as ** f of g ** and can be written as ** fog(x) **

Similarly, g(f(x)) is read as ** g of f ** and can be written as ** gof(x) **

Domain and Range
The composite function f( g(x) ) is only defined if the __range__ of g(x) is equal to or a subset of the __domain__ of f(x)

The output from g(x) has to become the input to f(x) so it has to ‘fit’.

The __ domain __ of f( g(x) ) is the domain of g(x)

The __ range __ of f( g(x) ) will be the output from ** f ** given the input is the values coming out of ** g ** (ie the range of g)

** Example 2 **
math \text{If } f(x) = \sqrt{x} \qquad \text{Domain: } x \in [0, \; \infty) math

math \text{And } g(x) = x^3 \qquad \text{Range: } y \in R math

math f \big( g(x) \big) = \sqrt{ x^3 } math

The range of ** g ** is __not__ a subset of the domain of ** f ** (it doesn't fit)

So f( g(x) ) is __undefined__.

We could restrict the domain of g so that the range of g __does__ fit into the domain of f
 * BUT **

math \text{ie we want the range of g to be } [0, \; \infty) \text{ or smaller} math


 * Maximal Domain **

The ** maximal domain ** is the largest possible domain for g, which gives a range of g which does fit into the domain of f. {Sometimes the __**maximal domain**__ is referred to as the __**implied domain**__} In the example above, the largest possible domain for g (ie the ** maximal domain **) for which this is true is: math g(x) = x^3 \qquad \text{Domain: } x \in [0, \; \infty) \qquad \text{Range: } y \in [0, \; \infty) math

With this restricted domain, f( g(x)) __is__ defined. math f \big( g(x) \big) = \sqrt{ x^3 } \qquad x \in [0, \; \infty) math

The __ domain __ of f( g(x) ) is the same as the domain of g.

math x \in [0, \; \infty) \text{ is the } \textit{maximal domain} \text{ for which } f \big( g(x) \big) \text{ exists} math

** Example 3 **
If f(x) = sin(x) ... ... {domain: x Î R and range: y Î [–1, 1] } and g(x) = (x + ½) 2 ... ... {domain: x Î R and range: y Î [0, ¥ )}

Then ... ... g( f(x) ) = ( sin(x) + ½) 2

... ... Domain of gof(x) is domain of f ... { x Î R }

... ... gof(x) is defined because __range__ of f [–1, 1] “fits” into __domain__ of g.

... ... When [–1, 1] is used as the input for g, the output from g is y Î [0, 2¼]

So __ range __ of gof(x) is y Î [0, 2¼]

It sometimes helps to map the transformation of domain into range R —→ f(x) = sin(x) —→ [–1, 1] —→ g(x) = (x + ½) 2 —→ [0, 2¼]

Graphing Composite Functions
Graphing a composite function can be done using a variation on the Addition of Ordinates process.

Many composite graphs become complicated so graphing software is usually used.

** Example 4 **
Given f(x) = sin(x) and g(x) = (x + ½) 2

Use technology to sketch g(f(x))

Domain: x Î R Range: y Î [0, 2¼]

Questions
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