Part A: CompositionTables
x 
0 
1 
2 
3 
4 
5 
p(x) 
1 
0 
5 
2 
3 
4 
q(x) 
5 
2 
3 
1 
4 
8 
s(x) 






r(x) 






1. Determine p(4).
2. Determine q(p(4)).
3. Determine q(3).
4. Find p(q(3).
5. Let s(x) = q(p(x)). Fill in the row for s(x).
6. Let r(x) = p(q(x)). Fill in the row for r(x). What do
you notice about r(5)?
7. Does p(q(1)) = q(p(1))?
8. In general, do p(q(x)) and q(p(x)) have the same value ?
Part B: CompositionGraphs
1. Let f be the function whose graph below has the rising line; let g be
the function whose
graph has the falling line.
a. g(f(1)) = ______________
b. f(g(1)) = ______________
2. Use the graphs below to evaluate:
a) f(g(6)) = __________________________
b) g(f(2)) = __________________________
c) g(f(0)) = __________________________
Part C: CompositionFormulas
1. Use the words input and output, as appropriate, to fill in the
missing blanks:
The function f g t uses the ____________ of the function g as the
_____________ to
the function f. The function g f t uses the ____________ of the
function f as the
_____________ to the function g.
2. Let u x = p q x and v x = q p x where
p x = 3x  4 and q x = x^{2} + 5 .
a. Calculate u 4 and v 4 . Are they the same?
u 4 = _______
v 4 = _______
b. Find formulas for u x and v x in terms of x.
u x = ____________________
v x = ____________________
3. Let f x = x^{2} + 3 and g x = 2 x + 1.
a. f(7) = __________ b. g(3) = __________ c. f(g(3)) = __________
4. Let f (x) = 2 x + 1 and g (x) = x^{2}  5 . Determine a
formula for each of the following:
a) f(g(x))
b) g(f(x))
Part D: Decomposition
1. Consider the composite function
Find three functions (f, g, and h) such that
w (x) = f( g (h( x))) . [Do not use h x = x .]
h (x) = __________________________
g (x) = __________________________
f (x) = __________________________
2a. Now consider the composite function
Decompose f into three
functions, u, v, and w, such that f x = u v w x .
w x = _____________
v x = _____________
u x = _____________
2b. Can you see a way to decompose f into four
functions? Demonstrate how to do it:
3. Let
Decompose g into functions, f and h, such that g x = f h x .
h x = _____________
f x = _____________
MAT 150: Section 8.2: Inverse Functions
Part A: Determining the Inverse Function
METHOD #1
Use words to describe the function: first
forward, then how to undo the function,
then a formula. 
METHOD #2
Use algebra to solve for x . 
1. f (x) = 3x  7
forward:
undo:
formula: 
1. f (x) = 3x  7 
forward:
undo:
formula: 

forward:
undo:
formula: 

Part B: Verifying Inverses
Suppose f x = 2 x  4 and
Are f and g inverse functions?
a) Use algebraic methods to verify . That is, determine f g x and then
determine
g f x . If f g x = g f x , then the functions are inverses.
determine f(g(x):
determine g(f(x):
what do you conclude?
b) Demonstrate the inverse relationship by means of a
graph (Note: this lacks meaning in
contexts)
c) Explain verbally:
Describe in words what f “does to its input.”
i.
ii. 
Describe in words what g “does to its input.”
i.
ii. 
d) Fill in the cells for the output and then explain the
inverse relationship:
f x = 2 x  4 

input 
2 
3 
4 
5 
6 
7 
8 
output 











input 
0 
2 
4 
6 
8 
10 
12 
output 







Part C: Inverse Function Notation
1. Explain the difference in meaning of the notation f (2) = 5 versus
the notation f^{1}(5) = 2 .
2. Suppose the point (10, 5) lies on the graph of a function f. What do you
know about the
graph of finverse?
3. The number of people (in thousands) in a city is given by the function f(t) =
20 + 0.4t, where
t is the number of years since 1970.
a. In the context of this problem, explain what f(25) and f^{1}(25) mean (no calculations
required). What is the unit of measure (number of people or number of years) for
f(25)
and f^{1}(25) ?
b. Now calculate f ^{1}(25) .
4. The graph of f from problem 3 is shown below. Estimate f^{1}(25) by reading the graph
below.
5. Suppose we have the function w = j x where w represents the average daily
quantity of
water (in gallons) required by an oak tree of height x feet.
a. What does the expression j 25 represent ? What are its units of measure?
b. What does the expression j ^{1} 25 represent? What are its units of measure?
c. What does the following equation tell you about v: j v
= 50
d. Rewrite the statement j v = 50 in terms of j ^{1} .
e. On certain acreage, oak trees on average measure z feet high and an oak tree
of average
height requires p gallons of water. Represent this statement first in terms of j
and then
in terms of j^{1}.
MAT 150: Section 8.3: Operations on Functions
Year 
1997 
1998 
1999 
2000 
2001 
2002 
Crimes in Baytown 
793 
795 
807 
818 
825 
831 
Crimes in Sunland 
448 
500 
525 
566 
593 
652 







Baytown Population 
61,000 
62,100 
63,220 
64,350 
65,510 
66,690 
Sunland Population 
28,000 
28,588 
29,188 
29,801 
30,427 
31,066 
This table gives the number of violent crimes committed in two cities between
1997 and 2002. It also gives the population for these two cities.
1. What do you notice about the data? How do the cities compare ?
2. Find the per capita crime rate for each city in 1997. What operation did you
use?
3. Fill in the table for the per capita crime rate for both cities for years
1997  2002.
Year 
1997 
1998 
1999 
2000 
2001 
2002 
Per capita crime rate: Baytown 






Per capita crime rate: Sunland 






4. Based on your table, which city do you think is more dangerous? Why?
5. If P(t) represents the population of a city at time t and if C(t) represents
the crime rate at time t, define a new function R(t) to represent the per capita
crime rate. [Hint: write R(t) in terms of P(t) and C(t).]
6. Let f(x) = x + 1 and g(x) = 3x^{2}:
a. Determine f(x) + g(x)
b. Determine f(x)  g(x)
c. Determine f(x)g(x)
d. Determine f(x)/g(x)
7. Let f(x) = x + 5 and g(x) = x  5:
a. Determine f(x) + g(x)
b. Determine f(x)  g(x)
c. Determine f(x)g(x)
d. Determine f(x)/g(x)