NAME
1-5
DATE
PERIOD
Study Guide and Intervention
Parent Functions and Transformations
Parent Functions
A parent function is the simplest of the functions in a family.
Parent Function
Form
Notes
constant function
f(x) = c
graph is a horizontal line
identity function
f(x) = x
points on graph have coordinates (a, a)
quadratic function
f(x) = x2
graph is U-shaped
cubic function
f(x) = x3
graph is symmetric about the origin
square root function
f(x) =
graph is in first quadrant
√
x
1
f(x) = −
x
reciprocal function
graph has two branches
absolute value function
f(x) = | x |
graph is V-shaped
greatest integer function
f(x) = x
defined as the greatest integer less than
or equal to x; type of step function
The graph confirms that D = {x | x ∈ } and R = {y | y ∈ }.
y
The graph intersects the origin, so the x-intercept is 0 and
the y-intercept is 0.
It is symmetric about the origin and it is an odd function:
f (x) = x 3
0
x
f(-x) = -f(x).
The graph is continuous because it can be traced without
lifting the pencil off the paper.
As x decreases, y approaches negative infinity, and as
x increases, y approaches positive infinity.
lim f(x) = -∞ and lim f(x) = ∞
x → -∞
x→∞
The graph is always increasing, so it is increasing for (-∞, ∞).
Exercise
Describe the following characteristics of the graph of the parent function
f(x) = x2 : domain, range, intercepts, symmetry, continuity, end behavior,
and intervals on which the graph is increasing/decreasing.
Chapter 1
27
Glencoe Precalculus
Lesson 1-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example
Describe the following characteristics of the graph of
the parent function f(x) = x3 : domain, range, intercepts, symmetry,
continuity, end behavior, and intervals on which the graph is
increasing/decreasing.
NAME
DATE
1-5
PERIOD
Study Guide and Intervention
(continued)
Parent Functions and Transformations
Transformations of Parent Functions
Parent functions can be transformed to
create other members in a family of graphs.
g(x) = f(x) + k is the graph
of f(x) translated…
…k units up when k > 0.
g(x) = f(x - h) is the graph
of f(x) translated…
…h units right when h > 0.
…k units down when k < 0.
Translations
…h units left when h < 0.
g(x) = -f(x) is the graph
of f(x)…
…reflected in the x-axis.
g(x) = f(-x) is the graph
of f(x)…
…reflected in the y-axis.
Reflections
g(x) = a f(x) is the graph
of f(x)…
…expanded vertically if a > 1.
g(x) = f(ax) is the graph
of f(x)…
…compressed horizontally if a > 1.
…compressed vertically if 0 < a < 1.
Dilations
…expanded horizontally if 0 < a < 1.
Example
Identify the parent function f(x) of g(x) = √
-x - 1, and describe how
the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.
y
g(x) = √-x - 1
f(x) = √x
0
x
Exercises
Identify the parent function f(x) of g(x), and describe how the graphs of g(x) and
f(x) are related. Then graph f(x) and g(x) on the same axes.
1. g(x) = 0.5 ⎪x + 4⎥
8
2. g(x) = 2x2 - 4
y
8
4
4
−8
−4 0
Chapter 1
y
4
8x
−8
4
−4
−4
−4
−8
−8
28
8x
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The graph of g(x) is the graph of the square root
function f(x) = √
x reflected in the y-axis and
then translated one unit down.
NAME
DATE
1-5
PERIOD
Practice
Parent Functions and Transformations
1. Use the graph of f(x) =
g(x) = √
x + 3 + 1.
√
x
to graph
2. Use the graph of f(x) = ⎪x⎥ to graph
g(x) = -|2x|.
y
y
x
0
0
x
y
3. Describe how the graph of f(x) = x2 and g(x) are
related. Then write an equation for g(x).
x
4. Identify the parent function f(x) of g(x) = 2|x + 2| - 3.
Describe how the graphs of g(x) and f(x) are related.
Then graph f(x) and g(x) on the same axes.
8
6
4
2
y
2 4 6 8x
−8−6−4
−4
−6
−8
⎨
⎩
-1 if x ≤ -3
5. Graph f(x) = 1 + x if -2 < x ≤ 2.
⎧ x if 4 ≤ x ≤ 6
6. Use the graph of f(x) = x3 to graph
g(x) = ⎪(x + 1)3⎥.
y
y
0
0
Chapter 1
x
x
Lesson 1-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
29
Glencoe Precalculus
NAME
1-5
DATE
PERIOD
Word Problem Practice
Parent Functions and Transformations
1. AREA The width w of a rectangular plot
of land with fixed area A is modeled by
A
the function w() = −
, where is the
length.
3. TAXES Graph the tax rates for the
different incomes by using a step
function.
Income Tax Rates for a Couple
Filing Jointly
a. If the area is 1000 square feet,
describe the transformations of the
1
parent function f(x) = −
x used to graph
w(x).
Limits of Taxable
Income ($)
b. Describe a function of the length that
could be used to find a minimum
perimeter for a given area
Tax Rate
(%)
0 to 41,200
15
41,201 to 99,600
28
99,601 to 151,750
31
151,751 to 271,050
36
271,051 and up
39.6
Source: Information Please Almanac
50
Tax Rate (%)
c. Is the function you found in part b a
transformation of f(x)? Explain.
40
30
20
10
d. Find the minimum perimeter for an
area of 1000 square feet.
30
90
120 150 180 210 240 270 300
4. HORIZON The function f(x) = √1.5x
can be used to approximate the distance
to the apparent horizon, or how far a
person can see on a clear day, where
f(x) is the distance in miles and x is the
person’s elevation in feet.
1 2
x + 2x,
can be modeled by h(x) = - −
10
where h(x) is the distance above the
ground in yards and x is the horizontal
distance from the tee in yards.
a. Describe the transformation of the
parent function f(x) = x2 used to graph
h(x).
a. How does the graph of f(x) compare
to the graph of its parent function?
b. The function g(x) = 1.2 √
x is also
used to approximate the distance to
the horizon. How does the graph of
g(x) compare to the graph of its
parent function?
b. Suppose the same shot was made
from a tee located 10 yards behind the
original tee. Rewrite h(x) to reflect
this change.
30
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Taxable Income
(thousands)
2. GOLF The path of the flight of a golf ball
Chapter 1
60
NAME
1-6
DATE
PERIOD
Study Guide and Intervention
Function Operations and Composition of Functions
Operations with Functions
Two functions can be added, subtracted,
multiplied, or divided to form a new function. For the new function, the
domain consists of the intersection of the domains of the two functions,
excluding values that make a denominator equal to zero.
Example 1
Given f(x) = x2 - x - 6 and g(x) = x + 2, find each
function and its domain.
f
a. (f + g)(x)
b. −
g (x)
(f + g)x = f(x) + g(x)
f
f(x)
−
x=−
2
g
g(x)
=x -x-6+x+2
2
x
-x-6
= −
= x2 - 4
()
()
x+2
(x - 3)(x + 2)
= − =x-3
x+2
The domains of f and g are both
(-∞, ∞), but x = -2 yields a zero in
The domains of f and g are both
(-∞, ∞), so the domain of (f + g) is
(-∞, ∞).
(f)
the denominator of −
g . So, the domain
is {x | x ≠ -2, x ∈ }.
Example 2
1
Given f(x) = x2 - 3 and g(x) = −
, find each function and its domain.
x
a. (f - g)(x)
b. (f g)(x)
(f - g)x = f(x) - g(x)
1
= x2 - 3 - −
(f g)x = f(x) g(x)
3
=x-−
x
x
The domain of f is (-∞, ∞) and the
domain of g is (−∞, 0) ∪ (0, ∞), so the
domain of (f - g) is (−∞, 0) ∪ (0, ∞).
The domain of f is (-∞, ∞) and the
domain of g is (−∞, 0) ∪ (0, ∞), so the
domain of (f - g) is (−∞, 0) ∪ (0, ∞).
Exercises
()
f
Find (f + g)(x), (f - g)(x), (f g)(x), and − (x) for each f(x) and g(x).
g
State the domain of each new function.
2
1. f(x) = x2 - 1, g(x) = −
x
Chapter 1
2. f(x) = x2 + 4x − 7, g(x) =
32
√
x
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
= (x2 - 3) −
x
NAME
DATE
1-6
Study Guide and Intervention
PERIOD
(continued)
Compositions of Functions
In a function composition, the result of
one function is used to evaluate a second function.
Given functions f and g, the composite function f ◦ g can be described by the
equation [f ◦ g](x) = f[g(x)]. The domain of f ◦ g includes all x-values in the
domain of g for which g(x) is in the domain of f.
Example
Given f(x) = 3x2 + 2x - 1 and g(x) = 4x + 2, find [f ◦ g](x)
and [g ◦ f](x).
[f ◦ g](x) = f[g(x)]
Definition of composite functions
= f(4x + 2)
Replace g(x) with 4x + 2.
= 3(4x + 2)2 + 2(4x + 2) - 1
Substitute 4x + 2 for x in f(x).
= 3(16x2 + 16x + 4) + 8x + 4 - 1
Simplify.
= 48x2 + 56x + 15
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
[g ◦ f](x) = g(f(x))
Definition of composite functions
= g(3x2 + 2x - 1)
Replace f(x) with 3x2 + 2x - 1.
= 4(3x2 + 2x - 1) + 2
Substitute 3x2 + 2x - 1 for x in g(x).
= 12x2 + 8x - 2
Simplify.
Exercises
For each pair of functions, find [f ◦ g](x), [g ◦ f](x), and [f ◦ g](4).
1. f(x) = 2x + 1, g(x) = x2 - 2x - 4
1
2. f(x) = 3x2 − 4, g(x) = −
x
3. f(x) = x3, g(x) = 5x
4. f(x) = 4x − 2, g(x) = √
x+3
5. f(x) = 3x - 5, g(x) = x2 + 1
1
6. f(x) = −
, g(x) = x2 - 1
1
7. f(x) = 2x - 3, g(x) = −
8. f(x) = x - 8, g(x) = x + 4
x-2
Chapter 1
x-1
33
Glencoe Precalculus
Lesson 1-6
Function Operations and Composition of Functions
NAME
1-6
DATE
PERIOD
Practice
Function Operations and Composition of Functions
(f)
Find (f + g)(x), (f - g)(x), (f · g)(x), and −
g (x) for each f(x) and
g(x). State the domain of each new function.
1. f(x) = 2x2 + 8 and g(x) = 5x - 6
2. f(x) = x3 and g(x) = √
x+1
For each pair of functions, find [f ◦ g](x), [g ◦ f](x), and [f ◦ g](3).
3. f(x) = x + 5 and g(x) = x - 3
4. f(x) = 2x3 - 3x2 + 1 and g(x) = 3x
5. f(x) = 2x2 - 5x + 1 and g(x) = 2x - 3
6. f(x) = 3x2 - 2x + 5 and g(x) = 2x - 1
7. f(x) = √x
-2
1
8. f(x) = −
x-8
g(x) = x2 + 5
g(x) = 3x
Find two functions f and g such that h(x) = [f ◦ g](x). Neither
function may be the identity function f(x) = x.
9. h(x) = √
2x - 6 -1
1
10. h(x) = −
3x +3
11. RESTAURANT A group of three restaurant patrons order the same meal
and drink and leave an 18% tip. Determine functions that represent the
cost of all of the meals before tip, the actual tip, and the composition of
the two functions that gives the cost for all of the meals including tip.
Chapter 1
34
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find f ◦ g.
NAME
1-6
DATE
PERIOD
Word Problem Practice
1. MARCHING BAND Band members
form a circle of radius r when the music
starts. They march outward as they
play. The function f(t) = 2.5t gives the
radius of
the circle in feet after t seconds.
Using g(r) = πr2 for the area of the
circle, write a composite function that
gives the area of the circle after t
seconds.
Then find the area, to the nearest tenth,
after 4 seconds.
4. TRAVEL Two travelers are budgeting
money for the same trip. The first
traveler’s budget (in dollars) can be
represented by f(x) = 45x + 350. The
second traveler’s budget (in dollars) can
be represented by g(x) = 60x + 475, x is
the number of nights.
a. Find (f + g)(x) and the relevant
domain.
b. What does the composite function in
part a represent?
2. CANDLES A hobbyist makes and sells
candles at a local market. The function
c(h) = 4h gives the number of candles
she has made after h hours. The function
f(c) = 12 + 0.25c gives the cost of making
c candles.
c. Find (f + g)(7) and explain what the
value represents.
d. Repeat parts a–c for (g - f)(x).
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Write the composite function that
gives the cost of candle making after
h hours.
b. A sale reduces the cost of making
c candles by 10%. Write the sale
function s(x) and the composite
function that gives the cost of candle
making after h hours if materials are
purchased during the sale.
5. POPULATION The function
p(x) = 2x2 - 12x + 18 predicts the
population of elk in a forest for the years
2010 through 2015 where x is the
number of years since 2000. Decompose
the function into two separate functions,
a(x) and b(x), so that [a ◦ b](x) = p(x) and
a(x) is a quadratic function and b(x) is a
linear function.
√
2x
28
3. SCIENCE The function t(x) = − + 6.25
gives the temperature in degrees Celsius
of the liquid in a beaker after x seconds.
Decompose the function into two
separate functions, s(x) and r(x), so that
s(r(x)) = t(x).
Chapter 1
35
Glencoe Precalculus
Lesson 1-6
Function Operations and Composition of Functions
TI-Nspire Activity
PERIOD
A12
2
Glencoe Precalculus
005_026_PCCRMC01_893802.indd 26
Chapter 1
-9
7. f(x) = x4 - 3x3 - x2 - 6; [1, 2]
11
5. f(x) = x4 - x3 + 7x; [0, 2]
16
3. f(x) = x + 7x - 11; [4, 5]
17
-−
2
1. f(x) = x3 + 4x2 - 6x - 5; [-4, -2]
4
3
26
6
8. f(x) = x2 - 1; [1, 5]
-33
Glencoe Precalculus
6. f(x) = x4 - 3x3 - x2 - 6; [-2, -1]
-15
4. f(x) = x - x + 7x; [-2, -1]
-7
2. f(x) = x2 + 7x - 11; [-8, -6]
Use the method shown above to find the average rate of change of
each function on the given interval.
Exercises
Step 4: Press b and select MEASUREMENT > SLOPE. Choose the
line. The slope is -10, so the average rate of change for the
interval [-2, 0] is -10.
Step 3: Press b and choose POINTS & LINES > LINE. Connect the
points on the graph.
Step 2: Press b and choose POINTS & LINES >
POINT ON. Place two points anywhere on the graph.
Double-click on each x-coordinate, changing one to -2
and the other to 0. The y-coordinates will update. You
may need to adjust your viewing window to see the points.
Step 1: Add a GRAPHS & GEOMETRY page. Enter
the function rule in the function entry line.
Press / + G to hide the function entry line.
Example
For the function f(x) = x3 + 4x2 - 6x - 5,
find the average rate of change for the interval [-2, 0].
Given a function, you can
draw two points on the function, connect the points with a line, and then
find the slope of that line, giving you the average rate of change for that
interval.
DATE
3/22/09 5:51:32 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 1
Finding an Average Rate of Change
1-4
NAME
DATE
PERIOD
Notes
graph is symmetric about the origin
f(x) =
1
f(x) = −
x
f(x) = | x |
f(x) = x
square root function
reciprocal function
absolute value function
greatest integer function
x→∞
lim f(x) = -∞ and lim f(x) = ∞
0
y
x
f(x) = x 3
x → -∞
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
027-042_PCCRMC01_893802.indd 27
Chapter 1
27
decreasing for (-∞, 0) and increasing for (0, ∞)
x→∞
Lesson 1-5
3/22/09 5:51:42 PM
Glencoe Precalculus
to y-axis; even function; continuous; lim f(x) = ∞ and lim f(x) = ∞;
D = {x | x ∈ }, R = {y | y ≥ 0, y ∈ }; x-int: 0; y-int: 0; symmetric with respect
Describe the following characteristics of the graph of the parent function
f(x) = x2 : domain, range, intercepts, symmetry, continuity, end behavior,
and intervals on which the graph is increasing/decreasing.
Exercise
The graph is always increasing, so it is increasing for (-∞, ∞).
x → -∞
As x decreases, y approaches negative infinity, and as
x increases, y approaches positive infinity.
The graph is continuous because it can be traced without
lifting the pencil off the paper.
f(x) = -f(x).
It is symmetric about the origin and it is an odd function:
The graph intersects the origin, so the x-intercept is 0 and
the y-intercept is 0.
The graph confirms that D = {x | x ∈ } and R = {y | y ∈ }.
Example
Describe the following characteristics of the graph of
the parent function f(x) = x3 : domain, range, intercepts, symmetry,
continuity, end behavior, and intervals on which the graph is
increasing/decreasing.
defined as the greatest integer less than
or equal to x; type of step function
graph is V-shaped
graph has two branches
graph is in first quadrant
graph is U-shaped
f(x) = x3
cubic function
√
x
points on graph have coordinates (a, a)
f(x) = x
f(x) = x2
graph is a horizontal line
identity function
constant function
quadratic function
Form
f(x) = c
Parent Function
A parent function is the simplest of the functions in a family.
Parent Functions and Transformations
Study Guide and Intervention
Parent Functions
1-5
NAME
Answers (Lesson 1-4 and Lesson 1-5)
Dilations
Reflections
Translations
…expanded horizontally if 0 < a < 1.
…compressed horizontally if a > 1.
…compressed vertically if 0 < a < 1.
…expanded vertically if a > 1.
…reflected in the y-axis.
…reflected in the x-axis.
…h units left when h < 0.
…h units right when h > 0.
…k units down when k < 0.
…k units up when k > 0.
Parent functions can be transformed to
g(x) = √-x - 1
0
f(x) = √x
y
28
4
8x
Glencoe Precalculus
9/30/09 2:04:21 PM
Answers
Glencoe Precalculus
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
027-042_PCCRMC01_893802.indd 28
Chapter 1
y
The graph of g(x) is the graph of the
square function f(x) = x2 expanded
vertically and translated 4 units down.
−8
−8 −4
−8
8x
−4
4
4
8
2. g(x) = 2x - 4
−4
−4 0
y
The graph of g(x) is the graph of the
absolute value function f(x) = |x|
compressed vertically and translated
4 units left.
−8
4
8
1. g(x) = 0.5 ⎪x + 4⎥
2
Identify the parent function f(x) of g(x), and describe how the graphs of g(x) and
f(x) are related. Then graph f(x) and g(x) on the same axes.
Exercises
The graph of g(x) is the graph of the square root
function f(x) = √
x reflected in the y-axis and
then translated one unit down.
Identify the parent function f(x) of g(x) = √
-x - 1, and describe how
the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.
g(x) = f(ax) is the graph
of f(x)…
g(x) = a f(x) is the graph
of f(x)…
g(x) = f(-x) is the graph
of f(x)…
g(x) = -f(x) is the graph
of f(x)…
g(x) = f(x - h) is the graph
of f(x) translated…
g(x) = f(x) + k is the graph
of f(x) translated…
PERIOD
(continued)
Parent Functions and Transformations
create other members in a family of graphs.
Example
DATE
Study Guide and Intervention
Transformations of Parent Functions
1-5
NAME
DATE
f(x)
x
to graph
0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
027-042_PCCRMC01_893802.indd 29
Chapter 1
0
y
x
-1 if x ≤ -3
5. Graph f(x) = 1 + x if -2 < x ≤ 2.
x if 4 ≤ x ≤ 6
y
g(x)
f(x)
x
g(x)
−8−6−4
−4
−6
−8
8
6
4
2
0
y
y
29
g(x)
0
y
x
f(x)
x
2 4 6 8x
g(x)
Lesson 1-5
10/23/09 4:24:52 PM
Glencoe Precalculus
6. Use the graph of f(x) = x3 to graph
g(x) = ⎪(x + 1)3⎥.
The graph of g(x) is the graph of f(x) = | x |
stretched vertically and translated 2 units left
and 3 units down.
4. Identify the parent function f(x) of g(x) = 2|x + 2| - 3.
Describe how the graphs of g(x) and f(x) are related.
Then graph f(x) and g(x) on the same axes.
g(x) is f(x) reflected in the x-axis,
translated 1 unit right and 1 unit up.
g(x) = -(x - 1)2 + 1
PERIOD
2. Use the graph of f(x) = ⎪x⎥ to graph
g(x) = -|2x|.
3. Describe how the graph of f(x) = x2 and g(x) are
related. Then write an equation for g(x).
0
y g(x)
√
x
Parent Functions and Transformations
Practice
1. Use the graph of f(x) =
g(x) = √
x + 3 + 1.
1-5
NAME
⎧
A13
⎨
Chapter 1
⎩
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 1-5)
A14
Glencoe Precalculus
027-042_PCCRMC01_893802.indd 30
Chapter 1
1 2
h(x) = - −
x + 10
10
b. Suppose the same shot was made
from a tee located 10 yards behind the
original tee. Rewrite h(x) to reflect
this change.
h(x) is the graph of f(x)
translated 10 units right,
compressed vertically,
reflected in the x-axis, and then
translated 10 units up.
a. Describe the transformation of the
parent function f(x) = x2 used to graph
h(x).
1 2
x + 2x,
can be modeled by h(x) = - −
10
where h(x) is the distance above the
ground in yards and x is the horizontal
distance from the tee in yards.
2. GOLF The path of the flight of a golf ball
d. Find the minimum perimeter for an
area of 1000 square feet. 126.5 ft
sample answer: they are two
different kinds of rational
functions.
c. Is the function you found in part b a
transformation of f(x)? Explain. No;
A
P(ℓ) = 2ℓ + 2 −
(ℓ)
b. Describe a function of the length that
could be used to find a minimum
perimeter for a given area
a. If the area is 1000 square feet,
describe the transformations of the
1
parent function f(x) = −
x used to graph
w(x). f(x) is expanded vertically.
30
PERIOD
36
39.6
151,751 to 271,050
271,051 and up
30
60
Taxable Income
(thousands)
90 120 150 180 210 240 270 300
Glencoe Precalculus
It is the parent function
expanded vertically.
b. The function g(x) = 1.2 √
x is also
used to approximate the distance to
the horizon. How does the graph of
g(x) compare to the graph of its
parent function?
It is the parent function
compressed horizontally.
a. How does the graph of f(x) compare
to the graph of its parent function?
4. HORIZON The function f(x) = √1.5x
can be used to approximate the distance
to the apparent horizon, or how far a
person can see on a clear day, where
f(x) is the distance in miles and x is the
person’s elevation in feet.
10
20
30
40
50
31
99,601 to 151,750
Source: Information Please Almanac
15
28
41,201 to 99,600
Tax Rate
(%)
0 to 41,200
Limits of Taxable
Income ($)
Income Tax Rates for a Couple
Filing Jointly
3. TAXES Graph the tax rates for the
different incomes by using a step
function.
Parent Functions and Transformations
Word Problem Practice
1. AREA The width w of a rectangular plot
of land with fixed area A is modeled by
A
the function w() = −
, where is the
length.
1-5
DATE
3/22/09 5:52:06 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 1
Tax Rate (%)
NAME
Enrichment
DATE
PERIOD
"
"'
0
y
y
8x
−8
−4
4
8
−8
−4
2
y
4
1 2
5. f(x) = −
x -1
8x
y
4
8x
−8
−8
−4
−4 0
4
8
y
4
7. f(x) = √
-x - 4
8x
3
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
027-042_PCCRMC01_893802.indd 31
Chapter 1
31
represents the rotated graph? f(x) = - − x + −
1
2
8. The graph of the function f(x) = 2x - 3 is rotated 90°. What function
−8
−4
−8 −4 0
4
6. f(x) = ⎪x3 + 2x - 4⎥
"
"
x
Lesson 1-5
3/22/09 5:52:11 PM
Glencoe Precalculus
Graph each function. Then graph the function after it is rotated 270°.
−8
−8 −4 0
4
8
4. f(x) = x3 - 2
Graph each function. Then graph the function after it is rotated 90°.
To rotate a function, you can plot several image points and then connect them.
3. Rotate point A'' by 90°. Graph the point. Give the
coordinates of A'''. Then use the result to write a rule for
rotating (x, y) by 270°. (2, -3); (x, y) → (y, -x)
2. Rotate point A' by 90°. Graph the point. Give the
coordinates of A''. Then use the result to write a rule for
rotating (x, y) by 180°. (-3, -2); (x, y) → (-x, -y)
1. Rotate point A by 90° using the rule. Graph the point.
Give the coordinates of A'. (-2, 3)
A rotation is a rigid transformation. A rotation turns a figure about a point
a certain number of degrees. The rotation can be clockwise or
counterclockwise. For this activity, assume all rotations are about the origin
and in the counterclockwise direction. To rotate a point 90° about the origin,
use the rule (x, y) → (-y, x).
Rotations
1-5
NAME
Answers (Lesson 1-5)
Chapter 1
DATE
PERIOD
Function Operations and Composition of Functions
Study Guide and Intervention
A15
f
()
x
The domain of f is (-∞, ∞) and the
domain of g is (−∞, 0) ∪ (0, ∞), so the
domain of (f - g) is (−∞, 0) ∪ (0, ∞).
x
3
=x-−
1
= (x2 - 3) −
(f g)x = f(x) g(x)
x
32
√
x
Glencoe Precalculus
9/30/09 2:04:35 PM
Answers
Glencoe Precalculus
x2 + 4x - 7
− ; D = (0, ∞)
√
x
x2 √
x + 4x √
x - 7 √
x ; D = [0, ∞)
x2 + 4x - 7 − √
x ; D = [0, ∞)
x + 4x - 7 + √
x ; D = [0, ∞)
2
2. f(x) = x2 + 4x − 7, g(x) =
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
027-042_PCCRMC01_893802.indd 32
Chapter 1
x -x+2
−
; D = (-∞, 0) ∪ (0, ∞)
x
x3 - x - 2
−
; D = (-∞, 0) ∪ (0, ∞)
x
2x2 - 2
−
;
D
= (-∞, 0) ∪ (0, ∞)
x
x3 - x
−; D = (-∞, 0) ∪ (0, ∞)
2
3
2
1. f(x) = x2 - 1, g(x) = −
Find (f + g)(x), (f - g)(x), (f g)(x), and − (x) for each f(x) and g(x).
g
State the domain of each new function.
Exercises
The domain of f is (-∞, ∞) and the
domain of g is (−∞, 0) ∪ (0, ∞), so the
domain of (f - g) is (−∞, 0) ∪ (0, ∞).
(f - g)x = f(x) - g(x)
1
= x2 - 3 - −
x
b. (f g)(x)
a. (f - g)(x)
x
1
Given f(x) = x2 - 3 and g(x) = −
, find each function and its domain.
is {x | x ≠ -2, x ∈ }.
the denominator of −
g . So, the domain
(f)
The domains of f and g are both
(-∞, ∞), but x = -2 yields a zero in
= − =x-3
x+2
(x - 3)(x + 2)
x+2
2
-x-6
= x−
()
f(x)
(−gf )x = −
g(x)
Example 2
The domains of f and g are both
(-∞, ∞), so the domain of (f + g) is
(-∞, ∞).
= x2 - 4
(f + g)x = f(x) + g(x)
= x2 - x - 6 + x + 2
Example 1
Given f(x) = x2 - x - 6 and g(x) = x + 2, find each
function and its domain.
f
b. −
a. (f + g)(x)
g (x)
Two functions can be added, subtracted,
multiplied, or divided to form a new function. For the new function, the
domain consists of the intersection of the domains of the two functions,
excluding values that make a denominator equal to zero.
Operations with Functions
1-6
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
(continued)
PERIOD
Function Operations and Composition of Functions
Study Guide and Intervention
Simplify.
= 3(16x2 + 16x + 4) + 8x + 4 - 1
= 12x + 8x - 2
Simplify.
Substitute 3x2 + 2x - 1 for x in g(x).
Replace f(x) with 3x2 + 2x - 1.
x-2
2x - 5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
027-042_PCCRMC01_893802.indd 33
Chapter 1
8 - 3x
1
−
;−
; -2
x-2
1
7. f(x) = 2x - 3, g(x) = −
3x2 - 2; 9x2 - 30x + 26; 46
5. f(x) = 3x - 5, g(x) = x2 + 1
125x3; 5x3; 8000
3. f(x) = x3, g(x) = 5x
2x2 - 4x - 7; 4x2 - 5; 9
1. f(x) = 2x + 1, g(x) = x2 - 2x - 4
3x - 4
16
2
33
x - 4; x - 4; 0
8. f(x) = x - 8, g(x) = x + 4
x - 2 x - 2x + 1 14
1
2x - x
1
−
;−
;−
2
2
x-1
1
6. f(x) = −
, g(x) = x2 - 1
Lesson 1-6
3/22/09 5:52:26 PM
Glencoe Precalculus
-2
4 √
x + 3 - 2; √
4x + 1 ; 4 √7
4. f(x) = 4x − 2, g(x) = √
x+3
x
3 - 4x
61
1
−
;−
; -−
2
2
2
1
2. f(x) = 3x2 − 4, g(x) = −
x
For each pair of functions, find [f ◦ g](x), [g ◦ f](x), and [f ◦ g](4).
Exercises
2
= 4(3x + 2x - 1) + 2
2
= g(3x2 + 2x - 1)
[g ◦ f](x) = g(f(x))
Definition of composite functions
Substitute 4x + 2 for x in f(x).
= 3(4x + 2)2 + 2(4x + 2) - 1
= 48x2 + 56x + 15
Replace g(x) with 4x + 2.
= f(4x + 2)
Definition of composite functions
Given f(x) = 3x2 + 2x - 1 and g(x) = 4x + 2, find [f ◦ g](x)
[f ◦ g](x) = f[g(x)]
and [g ◦ f](x).
Example
Given functions f and g, the composite function f ◦ g can be described by the
equation [f ◦ g](x) = f[g(x)]. The domain of f ◦ g includes all x-values in the
domain of g for which g(x) is in the domain of f.
In a function composition, the result of
one function is used to evaluate a second function.
Compositions of Functions
1-6
NAME
Answers (Lesson 1-6)
(f)
2x2 + 5x + 2, D = (-∞, ∞)
1. f(x) = 2x2 + 8 and g(x) = 5x - 6
2
{|
5
}
3
x
−
, D = (-1, ∞)
√
x+1
3
x √
x + 1 , D = [-1, ∞)
x3 - √
x + 1 , D = [-1, ∞)
A16
2
g(x) = x + 1
1
Sample answer: f(x) = −
,
3x
3x +3
1
10. h(x) = −
Glencoe Precalculus
027-042_PCCRMC01_893802.indd 34
Chapter 1
34
Glencoe Precalculus
f(x) = 3x, where x is the cost for one meal; g(x) = 1.18x; g(f(x)) = 3.54x
11. RESTAURANT A group of three restaurant patrons order the same meal
and drink and leave an 18% tip. Determine functions that represent the
cost of all of the meals before tip, the actual tip, and the composition of
the two functions that gives the cost for all of the meals including tip.
Sample answer: f(x) = √
x - 1,
g(x) = 2x - 6
9. h(x) = √2x
- 6 -1
Find two functions f and g such that h(x) = [f ◦ g](x). Neither
function may be the identity function f(x) = x.
1
{x | x ≠ ± √3, x ∈ }; f ◦ g = −
x -3
3x - 2
{x | x ≥ −23 , x ∈ }; f ◦ g = √
x-8
g(x) = x2 + 5
1
8. f(x) = −
12x2 - 16x + 10; 6x2 - 4x + 9; 70
6. f(x) = 3x2 - 2x + 5 and g(x) = 2x - 1
54x3 - 27x2 + 1; 6x3 - 9x2 + 3; 1216
4. f(x) = 2x3 - 3x2 + 1 and g(x) = 3x
g(x) = 3x
7. f(x) = √x
-2
Find f ◦ g.
8x2 - 34x + 34; 4x2 - 10x - 1; 4
5. f(x) = 2x2 - 5x + 1 and g(x) = 2x - 3
x + 2; x + 2; 4
3. f(x) = x + 5 and g(x) = x - 3
For each pair of functions, find [f ◦ g](x), [g ◦ f](x), and [f ◦ g](3).
6
−, D = x x ≠ −
,x∈
2x2 + 8
5x - 6
10x - 12x + 40x - 48,
D = (-∞, ∞)
3
2x - 5x + 14, D = (-∞, ∞)
x3 + √
x + 1 , D = [-1, ∞)
2. f(x) = x3 and g(x) = √
x+1
g(x). State the domain of each new function.
2
PERIOD
Function Operations and Composition of Functions
Practice
DATE
9/30/09 2:04:49 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 1
Find (f + g)(x), (f - g)(x), (f · g)(x), and −
g (x) for each f(x) and
1-6
NAME
2
2
x
Sample answer: s(x) = −
+ 6.25;
28
2x
r(x) = √
gives the temperature in degrees Celsius
of the liquid in a beaker after x seconds.
Decompose the function into two
separate functions, s(x) and r(x), so that
s(r(x)) = t(x).
√
2x
28
3. SCIENCE The function t(x) = − + 6.25
s(x) = 0.9x;
s{f[c(h)]} = 10.8 + 0.9h
b. A sale reduces the cost of making
c candles by 10%. Write the sale
function s(x) and the composite
function that gives the cost of candle
making after h hours if materials are
purchased during the sale.
f[c(h)] = 12 + h
a. Write the composite function that
gives the cost of candle making after
h hours.
2. CANDLES A hobbyist makes and sells
candles at a local market. The function
c(h) = 4h gives the number of candles
she has made after h hours. The function
f(c) = 12 + 0.25c gives the cost of making
c candles.
g[f(t)] = 6.25πt ; 314.2 ft
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
027-042_PCCRMC01_893802.indd 35
Chapter 1
DATE
PERIOD
35
Lesson 1-6
10/1/09 10:15:58 PM
Glencoe Precalculus
Sample answer: a(x) = 2x2;
b(x) = x - 3
5. POPULATION The function
p(x) = 2x2 - 12x + 18 predicts the
population of elk in a forest for the years
2010 through 2015 where x is the
number of years since 2000. Decompose
the function into two separate functions,
a(x) and b(x), so that [a ◦ b](x) = p(x) and
a(x) is a quadratic function and b(x) is a
linear function.
b: how much more the second
traveler can spend than the
first
c: $230; how much more the
second traveler can spend on a
7-night trip
a: (g - f)(x) = 15x + 125;
D = {x | x ≥ 0, x ∈ }
d. Repeat parts a–c for (g - f)(x).
combined amount that can be
spent by the travelers on a
7-night trip
c. Find (f + g)(7) and explain what the
value represents. $1560; the
budget of both travelers
b. What does the composite function in
part a represent? the combined
D = {x | x ≥ 0, x ∈ }
a. Find (f + g)(x) and the relevant
domain. (f + g)(x) = 105x + 825;
4. TRAVEL Two travelers are budgeting
money for the same trip. The first
traveler’s budget (in dollars) can be
represented by f(x) = 45x + 350. The
second traveler’s budget (in dollars) can
be represented by g(x) = 60x + 475x is
the number of nights.
Function Operations and Composition of Functions
Word Problem Practice
1. MARCHING BAND Band members
form a circle of radius r when the music
starts. They march outward as they
play. The function f(t) = 2.5t gives the
radius of
the circle in feet after t seconds.
Using g(r) = πr2 for the area of the
circle, write a composite function that
gives the area of the circle after t
seconds.
Then find the area, to the nearest tenth,
after 4 seconds.
1-6
NAME
Answers (Lesson 1-6)