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)