HW 1.2.4: Composite Functions
Given each pair of functions, calculate f  g  0   and g  f  0   .
1. f  x   4 x  8 , g  x   7  x2
2. f  x   5x  7 , g  x   4  2x2
3. f  x   x  4 , g  x   12  x3
4. f  x  
1
, g  x   4x  3
x2
Use the table of values to evaluate each expression
5.
f ( g (8))
6.
f  g  5 
7. g ( f (5))
8. g  f  3 
9.
f ( f (4))
10. f  f 1 
11. g ( g (2))
x
0
1
2
3
4
5
6
7
8
9
f ( x)
7
6
5
8
4
0
2
1
9
3
g( x)
9
5
6
2
1
8
7
3
4
0
12. g  g  6  
Use the graphs to evaluate the expressions below.
13. f ( g (3))
14. f  g 1 
15. g ( f (1))
16. g  f  0  
17. f ( f (5))
18. f  f  4  
19. g ( g (2))
20. g  g  0  
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
For each pair of functions, find f  g  x   and g  f  x   . Simplify your answers.
21. f  x  
1
7
, g  x   6
x6
x
22. f  x  
1
2
, g  x   4
x4
x
23. f  x   x2  1 , g  x   x  2
24. f  x   x  2 , g  x   x2  3
25. f  x   x , g  x   5x  1
26. f  x   3 x , g  x  
x 1
x3
27. If f  x   x4  6 , g ( x)  x  6 and h( x)  x , find f ( g (h( x)))
28. If f  x   x 2  1, g  x  
1
and h  x   x  3 , find f ( g (h( x)))
x
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
29. Given functions p  x  
interval notation.
a. Domain of
1
and m  x   x 2  4 , state the domains of the following functions using
x
p  x
m  x
b. Domain of p(m( x))
c. Domain of m( p ( x ))
30. Given functions q  x  
interval notation.
a. Domain of
1
and h  x   x 2  9 , state the domains of the following functions using
x
q  x
h  x
b. Domain of q (h( x))
c. Domain of h(q( x))
31. The function D ( p ) gives the number of items that will be demanded when the price is p. The
production 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
32. The function A(d ) gives the pain level on 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 at a pain level of 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
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
33. The radius r, in inches, of a spherical balloon is related to the volume, V, by r (V )  3
pumped into the balloon, so the volume after t seconds is given by V  t   10  20t .
3V
. Air is
4
a. Find the composite function r V  t  
b. Find the time when the radius reaches 10 inches.
34. The number of bacteria in a refrigerated food product is given by N T   23T 2  56T 1 ,
3  T  33 , where T is the temperature of the food. When the food is removed from the
refrigerator, the temperature is given by T (t )  5t  1.5 , where t is the time in hours.
a. Find the composite function N T  t  
b. Find the time when the bacteria count reaches 6752
Find functions f ( x) and g ( x ) so the given function can be expressed as h  x   f  g  x   .
35. h  x    x  2 
37. h  x  
2
3
x5
39. h  x   3  x  2
36. h  x    x  5 
38. h  x  
3
4
 x  2
2
40. h  x   4  3 x
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
Selected Answers:
1. 𝑓(𝑔(0)) = 4(7) + 8 = 26, 𝑔(𝑓(0)) = 7 − (8)2 = −57
3. 𝑓(𝑔(0)) = √(12) + 4 = 4, 𝑔(𝑓(0)) = 12 − (2)3 = 4
5. 𝑓(𝑔(8)) = 4
7. 𝑔(𝑓(5)) = 9
9. 𝑓(𝑓(4)) = 4
11. 𝑔(𝑔(2)) = 7
13. 𝑓(𝑔(3)) = 0
15. 𝑔(𝑓(1)) = 4
17. 𝑓(𝑓(5)) = 3
19. 𝑔(𝑔(2)) = 2
21. 𝑓(𝑔(𝑥)) =
1
𝑥
= 7 , 𝑔(𝑓(𝑥)) =
7
( +6)−6
𝑥
7
(
1
)
𝑥−6
+ 6 = 7𝑥 − 36
23. 𝑓(𝑔(𝑥)) = (√𝑥 + 2)2 + 1 = 𝑥 + 3, 𝑔(𝑓(𝑥)) = √(𝑥 2 + 1) + 2 = √(𝑥 2 + 3)
25. 𝑓(𝑔(𝑥)) = |5𝑥 + 1|, 𝑔(𝑓(𝑥)) = 5|𝑥| + 1
4
27. 𝑓(𝑔(ℎ(𝑥))) = ((√𝑥) − 6) + 6
29. (a)
𝑝(𝑥)
𝑚(𝑥)
=
1
√𝑥
𝑥 2 −4
, which can be written as
1
. This function is undefined when there is a
√𝑥(𝑥 2 −4)
negative number under the square root, or when the factor in the denominator equals zero. So the
domain is all positive numbers excluding 2, or {𝑥|𝑥 > 0, 𝑥 ≠ 2}.
1
(b) 𝑝(𝑚(𝑥)) = √𝑥 2 . This function is undefined when there is a negative number under the
−4
square root or a zero in the denominator, which happens when 𝑥 is between -2 and 2. So the
domain is −2 > 𝑥 > 2.
1
(c) 𝑚(𝑝(𝑥)) = (
√
2
1
) − 4 = 𝑥 − 4. This function is undefined when the denominator is zero, or
𝑥
when 𝑥 = 0. So the domain is {𝑥|𝑥 ∊ ℝ, 𝑥 ≠ 0}.
31. b
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
3
3(10+20𝑡)
33. (a) 𝑟(𝑉(𝑡)) = √
4𝜋
3
3(10+20𝑡)
(b) Evaluating the function in (a) when 𝑟(𝑉(𝑡)) = 10 gives 10 = √
4𝜋
. Solving this for t
gives 𝑡 ≈ 208.93, which means that it takes approximately 208 seconds or 3.3 minutes to blow up a
balloon to a radius of 10 inches.
35. 𝑓(𝑥) = 𝑥 2 , 𝑔(𝑥) = 𝑥 + 2
3
37. 𝑓(𝑥) = 𝑥 , 𝑔(𝑥) = 𝑥 − 5
39. 𝑓(𝑥) = 3 + 𝑥, 𝑔(𝑥) = √𝑥 − 2
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/