FINDING COMPOSITION OF TWO FUNCTIONS WORKSHEET

Find the composition of the functions given below.

Problem 1 :

If f(x) = 3x - 5 and g(x) = x², find (f∘g) (x).

Solution

Problem 2 :

Let f(x) = -3x + 7 and g(x) = 2x² - 8, find (f∘g) (x) and (g∘f) (x).           Solution

Problem 3 :

 If f(x) = -9x - 9 and g(x) = √(x - 9), find (f∘g) (x).

Solution

Problem 4 :

If f(x) = -2x + 1 and g(x) = √(x² - 9), find

(i)  (f∘g) (x) and 

(ii)  (g∘f) (x)                   Solution

Problem 5 :

 If f(x) = -2x + 1 and g(x) = √(x² - 5), find

(i)  (f∘g) (x) and 

(ii)  (g∘f) (x)                 Solution

Problem 6 :

A banquet hall charges $975 to rent a reception room, plus $39.95 per person. Next month, however, the banquet hall will be offering a 20% discount off the total bill. Express this discounted cost as a function of the number of people attending.

Solution

Problem 7 :

For a car travelling at a constant speed of 80 km/hr, the distance driven d kilometers is represented by

d(t) = 80t

where t is is the time in hours. The cost of gasoline in dollars, for the drive is represented by 

C(d) = 0.09d

a) determine C(d(5)) numerically and interpret your result.

b)  Describe the relationship represented by C(d(t))

Solution

Problem 8 :

The function p(d) = 0.03d + 1 approximates the pressure (in atmospheres) at a depth of d feet below sea level. The function d(t) = 60t represents the depth (in feet) of a diver t minutes after beginning a descent from sea level, where 0 ≤ t ≤ 2.

a. Find p(d(t)). Interpret the terms and coefficient.

b. Evaluate p(d(1.5)) and explain what it represents

Solution

Problem 1 :

 Given f(x) = -9x + 3 and g(x) = x⁴, find (f ∘ g)(x)

Solution

Problem 2 :

Given f(x) = 2x – 5 and g(x) = x + 2, find (f ∘ g)(x)

Solution

Problem 3 :

Given f(x) = x² + 7 and g(x) = x - 3, find (f ∘ g)(x)

Solution

Problem 4 :

Given f(x) = 4x + 3 and g(x) = x², find (g ∘ f)(x)

Solution

Problem 5 :

Given f(x) = x – 1 and g(x) = x² + 2x - 8, find (g ∘ f)(x)

Solution

Problem 6 :

Use f(x) = 2x + 3 adn g(x) = 1 - x² to evaluate the following expressions.

a)  f (g(0))        b)  g (f(4))        c)  (f ∘ g) (-8)

d)  (g ∘ g) (1/2)     e)  (f ∘ f^-1) (1)      f)  (g ∘ g) (2)

Solution

Problem 7 :

Given f = {(0, 1) (1, 2) (2, 5) (3, 10)} and g = {(2, 0)(3, 1)(4, 2)(5, 3)(6, 4)} determine the follwoing

Solution

Composition of Functions with Square Roots

Problem 1 :

Let f(x) = x² - 1 and let g(x) = √x. Find the domain of the composition functions

(i)  g∘f 

(ii)  f∘g              Solution

Problem 2 :

 Let

f(x) = 1/(x² - 1) and g(x) = √(x - 2)

Find the domain of f(g(x))              Solution

Problem 3 :

Let

f(x) = √(x + 2) and g(x) = x²

Find the domain of f(g(x)).          Solution

Problem 4 :

The function D(p) gives the number of items that will be demanded when the price is p. The product cost C(x), is the cost of producing x items. To determine the cost of production when the price is $6, you would do which of the following.

a)  Evaluate D(C(6))      b)  Evaluate C(D(6))

c)  Solve D(C(x)) = 6      d)  Solve C(D(p)) = 6

Solution

Problem 5 :

The function A(d) gives the pain level of a scale of 0-10 experienced by a patient with d milligrams of a pain reduction drug in their system. The milligrams of drug in the patient's system after t minutes is modeled by m(t). To determine when the patient will be a pain level 4, you would need to 

a)  Evaluate A(m(4))      b)  Evaluate m(A(4))

c)  Solve A(m(t)) = 4      d)  Solve m(A(d)) = 4

Solution

Example 6 :

The radius r in inches of a spherical ballon is related to the Volume V, by 

r(V) = ∛(3V/4 π)

Air is pumped into a balloon, so the volume after t seconds is given by V(t) = 10 + 20t

a) Find the composite function r(V(t))

b)  Find the radius after 20 seconds.

Solution

Example 7 :

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to

r(t) = 25√(t + 2)

find the area of the ripple as a function of time. Find the area of the ripple at t = 2.

Solution

Example 8 :

For the following exercises, find the composition when

f(x) = x² + 2 for all x ≥ 0 and g(x) = √(x − 2)

a) (f ∘ g) (6), (g ∘ f) (6)

b) (g ∘ f) (a), (f ∘ g) (a)

c) (f ∘ g) (11), (g ∘ f) (11)

Solution