  Composition of Functions

Part A: Composition-Tables

 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: Composition-Graphs
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: Composition-Formulas
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 = x2 + 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 = x2 + 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) = x2 - 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 f-inverse?

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. Re-write 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) = 3x2:

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)

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