Composition of Functions

In addition to adding, subtracting, multplying and dividing, two functions can be composed. The composition of a function is when the x-value is replaced by a function. For example if p(x) = x3 and q(x) = x - 1, the compostition of p with q is:

pq=p( q( x ) )=p( x1 )= ( x1 ) 3



The notation pq , reads "p composed with q". Which means that the value of x is replaced by q(x) in function p.

THE DEFINITION OF COMPOSITION OF FUNCTIONS


The composition of the function p is

( pq )( x )=p( q( x ) )



where the domain of pq is the set of all x in the domain of q such that q(x) is in the domain of p.

Let's look at a couple examples.

Example 1: Given p(x) = x2 - 1 and q(x) = x + 3 find the composite function and its domain of ( pq )( x ) and ( qp )( x ) . Then evaluate ( pq )( 3 ).

Step 1: Find ( pq )( x ) and its domain.

( pq )( x )=p( q( x ) ) Def'n of compos.


p( x+3 )= ( x+3 ) 2 1  Sub. q(x)


( pq )= x 2 +6x+91 Simplify


( pq )= x 2 +6x+8


The domain of p(x) = x2 - 1 is the set of all real numbers.


The domain of q(x) = x + 3 is the set of all real numbers.


Therefore the domain of ( pq ) is the set of all real numbers.

Step 2: Find ( qp )( x ) and its domain.

( qp )( x )=q( p( x ) ) Def'n of compos.


q( x 2 1 )=( x 2 1 )+3 Sub. p(x)


( qp )= x 2 +2  Simplify


The domain of q(x) = x + 3 is the set of all real numbers.


The domain of p(x) = x2 - 1 is the set of all real numbers.


Therefore the domain of ( qp ) is the set of all real numbers.

Step 3: Evaluate ( pq )( x ) when x = 3.

( pq )( x )= x 2 +6x+8


( pq )( 3 )= 3 2 +6( 3 )+8=35

Example 2: Given p( x )= 4x and q(x) = x2 find the composite function and its domain of ( pq )( x ) and ( qp )( x ) . Then evaluate ( pq )( 5 ) .

Step 1: Find ( pq )( x ) and its domain.

( pq )( x )=p( q( x ) ) Def'n of compos.


p( x 2 )= 4 x 2     Sub. q(x)


( pq )=2x Simplify


The domain of p( x )= 4x is the set of all real numbers such that x0 .


The domain of q(x) = x2 is the set of all real numbers.


Therefore the domain of ( pq ) is the set of all real numbers such that x0 .

Step 2: Find ( qp )( x ) and its domain.

( qp )( x )=q( p( x ) )   Def'n of compos.


q( 4x )= ( 4x ) 2     Sub. p(x)


( qp )=4x   Simplify


The domain of q(x) = x2 is the set of all real numbers.


The domain of p( x )= 4x is the set of all real numbers such that x0 .


Therefore the domain of ( pq ) is the set of all real numbers such that x0 .

Step 3: Evaluate ( qp )( x ) when x = -5.

( pq )( x )=4x


( pq )( 5 )=4( 5 )=20





Related Links:
Math
algebra
Extreme Value Theorem
Even and Odd Functions
Calculus Topics


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