Copyright (c) 2014 Advanced Instructional Systems, Inc.
Given
2
( ) 2 1f x x
and
( ) 3 5g x x
, find the following.
a.
( (2))fg
b.
( ( ))f g x
c.
( ( ))g f x
d.
( )( )g g x
e.
( )( 2)ff
Solution
a. The notation
( (2))fg
, reads “ f of g of 2” and it means to find the value of the composition
of the functions when we input 2 into function g and then use that result as the input into
function f. To calculate the value of
( (2))fg
, the output of g(2) will be the input for
function f.
2
(2) 3(2) 5 1
(1) 2(1) 1 3
( (2)) 3
g
f
fg
b. The notation
( ( ))f g x
, reads “ f of g of x” and it means to find the composite function when
function f is evaluated at g(x). To find
( ( ))f g x
, we must evaluate the inside function g at x,
which produces g(x) and then evaluate the outside function f at g(x).
2
2
2
2
( ( )) (3 5)
2 3 5 1
2(9 30 25) 1
18 60 50 1
18 60 51
f g x f x
x
xx
xx
xx
c. The notation
( ( ))g f x
, reads “ g of f of x” and it means to find the composite function when
function g is evaluated at f (x).To find
( ( ))g f x
, we must evaluate the inside function f at x,
which produces f(x) and then evaluate the outside function g at f(x).
2
2
2
2
( ( )) 2 1
3 2 1 5
6 3 5
62
g f x g x
x
x
x
** note: looking at the answers to parts b and c, we can see that function composition is not
commutative.
Copyright (c) 2014 Advanced Instructional Systems, Inc.
d. The notation
( )( )g g x
, reads “ g of g of x”, it can also be written as
( ( ))ggx
and it means
to find the composite function when function g is evaluated at g (x).To find
( )( )g g x
, we
must evaluate the inside function g at x, which produces g(x) and then evaluate the outside
function g at g(x).
( )( ) 3 5
3 3 5 5
9 15 5
9 20
g g x g x
x
x
x
e. The notation
( )( 2)ff
, reads “ f of f of -2”, it can also be written as
( ( 2))ff
and it
means to find the value of the composite function when function f is evaluated at x=-2. To
calculate the value of
( )( 2)ff
, we must evaluate the inside function, f at x=-2 and then
evaluate the outside function f at the value of f(-2).
2
2
( 2) 2( 2) 1 2(4) 1 9
(9) 2(9) 1 2(81) 1 163
( 2) 163
f
f
ff
