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) = √(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  √(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 , + ∞)
condition b) is written as follows: 1  f(x)^{ 2} > 0
or x  1 > 0
or x < 1 or in interval form ( ∞ , 1)

The domain of g _{o} f is given by the intersection of the sets [2 , + ∞) and ( ∞ , 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) = √x. Find the composite function g(f(x)).
Solution to Question 8:

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

For x ≥ 0
g(f(x)) = √(x^{ 2}) =  x  = x

Hence g(f(x)) is defined as follows
g(f(x)) = x for x ≥ 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) = √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 , + ∞)
 [6 , + ∞)
 f o g) = {(2 , 6) , (4 , 7)}
 f o g = {(6 , 6) , (7 , 7) , (0 , 0)}
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