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Composition of Functions Help

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By — McGraw-Hill Professional
Updated on Aug 31, 2011

Introduction to Composition of Functions

Suppose that f and g are functions and that the domain of g contains the range of f . This means that if x is in the domain of f then f ( x ) makes sense but also g may be applied to f (x) (Fig. 1.46). The result of these two operations, one following the other, is called g composed with f or the composition of g with f . We write

Basics 1.8 Sets and Functions

Fig. 1.46

Examples

Example 1

Let f ( x ) = x 2 − 1 and g ( x ) = 3 x + 4. Calculate g ο f .

Solution 1

We have

Basics 1.8 Sets and Functions

Notice that we have started to work inside the parentheses: the first step was to substitute the definition of f , namely x 2 − 1, into our equation.

Now the definition of g says that we take g of any argument by multiplying that argument by 3 and then adding 4. In the present case we are applying g to x2 − 1. Therefore the right side of equation (*) equals

3 · ( x2 − 1) + 4.

This easily simplifies to 3x2 + 1. In conclusion,

g ο f ( x ) = 3x2 + 1.

Example 2

Let f ( t ) = ( t 2 − 2)/( t + 1) and g ( t ) = 2t + 1. Calculate g ο f and f ο g .

Solution 2

We calculate that

Basics 1.8 Sets and Functions

We compute g of any argument by doubling it and adding 1. Thus equation (**) equals

Basics 1.8 Sets and Functions

One of the main points of this example is to see that f ο g is different from g ο f . We compute f ο g :

Basics 1.8 Sets and Functions

So f ο g and g ο f are different functions.

You Try It: Let f ( x ) = ∣ x ∣ and Basics 1.8 Sets and Functions . Calculate f ο g ( x ) and g ο f ( x ).

We say a few words about recognizing compositions of functions.

Examples

Example 3

How can we write the function k ( x ) = (2 x + 3)2 as the composition of two functions g and f ?

Solution 3

Notice that the function k can be thought of as two operations applied in sequence. First we double and add three, then we square. Thus define f ( x ) = 2x + 3 and g ( x ) = x2 . Then k ( x ) = ( g ο f )( x ).

We can also compose three (or more) functions: Define

( h ο g ο f )( x ) = h ( g ( f ( x ))).

Example 4

Write the function k from the last example as the composition of three functions (instead of just two).

Solution 4

First we double, then we add 3, then we square. So let f(x) = 2x , g(x) = x + 3, h(x) = x2. Then k(x) = ( h ο g ο f )(x).

Example 5

Write the function

Basics 1.8 Sets and Functions

as the composition of two functions.

Solution 5

First we square t and add 3, then we divide 2 by the quantity just obtained. As a result, we define f ( t ) = t 2 + 3 and g (t) = 2/ t. It follows that r (t) = ( g ο f )(t).

You Try It: Express the function g ( x ) = 3/(x2 + 5) as the composition of two functions. Can you express it as the composition of three functions?

Practice problems for this concept can be found at: Calculus Basics Practice Test.

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