Questions on
composition of functions are presented and their detailed solutions discussed. These questions have been designed to help you deepen your understanding of the concept of composite functions as well as to develop the computational skills needed while solving questions related to these functions.
Question 1:
For f(x) = 2x + 3 and g(x) = x^{ 2} + 1, find the composite function defined by (f _{o} g)(x)
Solution to Question 1:

The definition of composite functions gives
(f _{o} g)(x) = f(g(x))
= 2 (g(x)) + 3
= 2( x^{ 2} + 1 ) + 3
=  2 x^{ 2} + 5
Question 2:
Given f(2) = 3, g(3) = 2, f(3) = 4 and g(2) = 5, evaluate
(f _{o} g)(3)
Solution to Question 2:

Use the definition of the composite function to write
(f _{o} g)(3) = f(g(3))

Substitute g(3) by its given value 2 and evaluate f(2)
(f _{o} g)(3) = f(2) = 3
Question 3:
Functions f and g are give by
f(x) = SQRT(x + 2)
and
g(x) = ln (1  x^{ 2})
Find the composite function defined by (g _{o} f)(x) and describe its domain.
Solution to Question 3:

Use the definition of the composite function to write
(g _{o} f)(x) = g(f(x))
= ln (1  f(x)^{ 2})
= ln (1  SQRT(x + 2)^{ 2})
= ln (1  (x + 2))
= ln ( x  1)

The domain of g _{o} f is the set of all values of x so that a) x is in the domain of f and b) f(x) is in the domain of g
condition a) is written as follows: x + 2 >= 0
or x >= 2 or in interval form [2 , + infinity)
condition b) is written as follows: 1  f(x)^{ 2} > 0
or x  1 > 0
or x < 1 or in interval form (infinity , 1)

The domain of g _{o} f is given by the intersection of the sets [2 , + infinity) and (infinity , 1) and is given by
[2 , 1)
Question 4:
Functions f and g are as sets of ordered pairs
f = {(2,1),(0,3),(4,5)}
and
g = {(1,1),(3,3),(7,9)}
Find the composite function defined by g _{o} f and describe its domain and range.
Solution to Question 4:

Use the definition of the composite function to find
(g _{o} f)(2) = g( f(2) ) = g(1) = 1
(g _{o} f)(0) = g( f(0) ) = g(3) = 3
(g _{o} f)(4) = g( f(4) ) = g(5) = undefined

Hence g _{o} f is given by
g _{o} f = { (2 , 1 ) ,(0 , 3) }

The domain d and range r of g _{o} f are given by
d = {2 , 0} and r = {1 , 3}
Question 5:
For f(x) = ln x, find the first derivative of the composite function defined by F(x) = (f _{o} f)(x)
Solution to Question 5:

The composite function F(x) is given by
F(x) = ln (ln (x))

Let u(x) = ln (x) so that F(x) is written
F(x) = ln (u(x))

We now use the chain rule to differentiate F(x)
F '(x) = [ d ln(u) / du ]* du / dx = [ 1 / u ] * [1 / x]
= 1 / [ x ln (x) ]
Question 6:
Write function F given below as the composition of two functions f and g
F(x) =  4 x^{ 2} + 2x  5 
Solution to Question 6:

One possibility is to write f and g as follows
f(x) =  x  and g(x)= 4 x^{ 2} + 2x  5

so that
F(x) = f(g(x)) = (f _{o} g)(x)
Question 7:
Write function F given below as the composition of two functions f and g, where g(x) = 1 / x and F(x) = (1 / x) / (1 + x).
Solution to Question 7:

If g(x) = 1 / x and F(x) = (1 / x) / (1 + x) and
f(x) = x / (1 + 1/x)

Then F(x) may be written as the composite function
F(x) = f(g(x)) = (f _{o} g)(x)
Question 8:
g(x) is a piecewise function defined by:
f(x) = x for x < 0
and
f(x) = x^{ 2} for x >= 0.
Function g is defined by g(x) = SQRT (x). Find the composite function g(f(x)).
Solution to Question 8:

For x < 0 we have
g(f(x)) = SQRT(x) and is not a real number.

For >= 0
g(f(x)) = SQRT(x^{ 2}) =  x  = x

Hence g(f(x)) is defined as follows
g(f(x)) = x for >= 0
Question 9:
True or False
f(g(x)) = g(f(x)) for any two functions f and g. Justify your answer.
Solution to Question 9:

False. Try f(x) = x + 1 and g(x) = x^{ 2}
f(g(x)) = x^{ 2} + 1
g(f(x)) = (x + 1) ^{ 2}

In general f(g(x)) and g(f(x)) are not equal.
Question 10:
Evaluate f(g(h(1))), if possible, given that
h(x) =   x  , g(x) = x  1 and f(x) = 1 / (x + 2)
Solution to Question 10:

We first find h(1)
h(1) =  1

We now find g(1)
g(1) = 2

Finally f(2) is undefined since division by zero is not allowed. Hence f(g(h(1))) is undefined and x = 1 is not in the domain of f(g(h(x)))
Exercises:

Evaluate f(g(3)) given that
f(x) =  x  6  + x^{ 2}  1 and g(x) = 2x
 Find f(x) and g(x) if the composite function
f(g(x)) = 2 sec(2x + 1)

Find the domain of the composite function
g _{o} f if f(x) = SQRT(x) and g(x) = 1 / x.

Find the range of the composite function f(g(x)) given that
f(x) = x + 4 and g(x) = x^{ 2} + 2
 Find the composite function (f o g)(x) given that
f = {(3,6) , (5,7) , (9,0)} and g = {(2,3) , (4,5) , (6,7)}
 Find the composite function (f o g)(x) given that
f = {(1,6) , (4,7) , (5,0)} and g = {(6,1) , (7,4) , (0,5)}
Answers to Above Exercises:
 35
 One possibility: f(x) = 2 sec (x) and g(x) = 2x + 1.
 [0 , 4) U (4 , +infinity)
 [6 , +infinity)
 f o g) = {(2 , 6) , (4 , 7)}
 f o g = {(6 , 6) , (7 , 7) , (0 , 0)}
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