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Section 10.1
The Algebra of Functions
Section 10.1Exercise #1
Chapter 10
2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x
a. f + g x
b. f – g x
c. fg x
d.
fx
g
a. f + g x
2 = + 16 + 4 – x x
= x2 – x + 20
2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x
2 = + 16 – 4 – x x
= x2 + 16 – 4 + x
b. f – g x
= x2 + x + 12
2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x
2 = + 16 4 – x x
= 4x2 – x3 + 64 – 16x
= – x3 + 4x2 – 16x + 64
c. fg x
2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x
d.
fx
g
=
x2 + 164 – x
, x 4
2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x
Section 10.1Exercise #3
Chapter 10
2If = + 2 and = + 3, find:f x x g x x
–. 2a g f
b. f g x
c. g f x
–. 2a g f
= – 2g f
2– 2 = – 2 + = 62 f
– 2 6= g f g = 6 + 3 = 9
2If = + 2 and = + 3, find:f x x g x x
= + 3f g x f x
2 = + 3 + 2x
= x2 + 6x + 9 + 2
= x2 + 6x + 11
b. f g x
2If = + 2 and = + 3, find:f x x g x x
2 = + 2g f x g x
2 = + 2 + 3x
= x2 + 5
c. g f x
2If = + 2 and = + 3, find:f x x g x x
Section 10.1Exercise #4
Chapter 10
fin– 1 3 + 1
If = and = ,5 2 –
d the domain 2
of
+ , – and .
xf x g x
x xf g f g fg
– 1 3 + 1 + = +
5a.
2 – 2 x
f g xx x
2 – 2 0x
2 2x
1x
D = 0, 1, is a real number x x x x
5 0x
0x
– 1 3 + 1 – = –
5b.
2 – 2 x
f g xx x
D = 0, 1, is a real number x x x x
fin– 1 3 + 1
If = and = ,5 2 –
d the domain 2
of
+ , – and .
xf x g x
x xf g f g fg
– 1 3 + 1
= – 2
c5
.2
xfg x
x x
D = 0, 1, is a real number x x x x
fin– 1 3 + 1
If = and = ,5 2 –
d the domain 2
of
+ , – and .
xf x g x
x xf g f g fg
OBJECTIVES
A Find the sum, difference, product, and quotient of two functions.
OBJECTIVES
B Find the composite of two functions.
OBJECTIVES
C Find the domain of (ƒ + g)(x), (ƒ – g)(x), (ƒg)(x), and
(ƒg)(x).
OBJECTIVES
D Solve an application.
DEFINITIONOPERATIONS WITH FUNCTIONS
0,ƒ(x)g(x)
ƒg
(ƒ + g)(x) = ƒ(x) + g(x)
(ƒ – g)(x) = ƒ(x) – g(x)
(ƒg)(x) = ƒ(x) g(x)
(x) = g(x) ( )
DEFINITIONCOMPOSITE FUNCTION
(ƒ o g)(x) = ƒ(g(x))
If ƒ and g are functions:
Section 10.2
Inverse Functions
OBJECTIVES
A Find the inverse of a function when the function is given as a set of ordered pairs.
OBJECTIVES
B Find the equation of the inverse of a function.
OBJECTIVES
C Graph a function and its inverse and determine whether the inverse is a function.
OBJECTIVES
D Solve applications involving functions.
DEFINITION
The relation obtained by reversing the order of x and y.
INVERSE OF A FUNCTION
FINDING THE EQUATION OF AN INVERSE FUNCTION
PROCEDURE
1.Interchange the roles of x and y.
2.Solve for y.
DEFINITION
If y = ƒ(x) is one-to-one, the inverse of ƒ is also a function, denoted by y = ƒ –1(x).
Section 10.2Exercise #6
Chapter 10
Let = 3,5 , 5,7 and , 7, :9 findS
a. The domain and range of S
b. S–1
c. The domain and range of S–1
d. The graph of S and S–1
Domain of = 3,5,7S
a. The domain and range of S
Range of = 5,7,9S
b. S–1
–1 = 5,3 , 7,5 , 9,7S
Let = 3,5 , 5,7 and , 7, :9 findS
c. The domain and range of S–1
–1Range of = 3,5,7S
–1Domain of = 5,7,9S
Let = 3,5 , 5,7 and , 7, :9 findS
d. The graph of S and S–1
y
x 10 5 0 0
5
10
S–1
S
Let = 3,5 , 5,7 and , 7, :9 findS
Section 10.2Exercise #8
Chapter 10
2 = = 3Find the inverse of . Is the inverse a function?f x y x
– 1 2: = 3f x x y
x3
= y2
y2 =
x3
y = ±
x3
= ± 3x3
The inverse is not a function.
Section 10.3
Exponential Functions
OBJECTIVES
A Graph exponential functions of the form ax or a –x (a > 0).
OBJECTIVES
B Determine whether an exponential function is increasing or decreasing.
OBJECTIVES
C Solve applications involving exponential functions.
DEFINITIONEXPONENTIAL FUNCTION
( ) ( 0 1)xƒ x = b b > , b
A function defined for all real values of x by:
DEFINITION
Increasing: rises left to right.
Decreasing: falls left to right.
INCREASING AND DECREASING FUNCTIONS
DEFINITIONNATURAL EXPONENTIAL FUNCTION, BASE e
ƒ(x) = ex
has the approximate value 2.178e
Section 10.3Exercise #9
Chapter 10
Graph y = 3x .
a. Is the inverse a function?
b. Is y = 3x increasing or decreasing?
Graph y = 3x .
a. Is the inverse a function?
x – 5 0 5
10 f x
Yesx y
0 1
1 3 –1
13
Graph y = 3x .
x – 5 0 5
10 f x
b. Is y = 3x increasing or decreasing?
increasing
Section 10.3Exercise #10
Chapter 10
A radioactive substance decays so that thenumber of grams present after t years is
G = 1000e–1.4x
Find, to the nearest gram, the amount of thesubstance present,
a. At the start. b. In 2 years.
When t = 0
a. At the start.
G = 1000e0
G = 1000
1000 grams are present at the start.
b. In 2 years.
When t = 2
G = 1000e 1.4 2 = 1000e–2.8
G = 1000 0.06081 = 60.81 = 61
61 grams are present in 2 years.
Section 10.4
Logarithmic Functions and their Properties
OBJECTIVES
A Graph logarithmic functions.
OBJECTIVES
B Write an exponential equation in logarithmic form and a logarithmic equation in exponential form.
OBJECTIVES
C Solve logarithmic equations.
OBJECTIVES
D Use the properties of logarithms to simplify logarithms of products, quotients, and powers.
OBJECTIVES
E Solve applications involving logarithmic functions.
DEFINITION
Means the exponent to which we raise 3 to get x.
LOG3x
DEFINITIONLOGARITHMIC FUNCTION
ƒ(x) = y = logbx
is equivalent to:
by = x (b > 0, b ≠ 1, and x > 0)
DEFINITIONEQUIVALENCE PROPERTY
For any b > 0, b ≠ 1,
bx = by
is equivalent to x = y.
DEFINITIONPROPERTIES OF LOGARITHMS
logbMN = logbM + logbN
logbMN
= logbM – logbN
logbMr = r logbM
DEFINITIONOTHER PROPERTIES OF LOGARITHMS
logb 1 = 0
logb b = 1
logb bx = x
Section 10.4Exercise #11
Chapter 10
Graph on the same coordinate axes.
a. f x = 2x
b. f x = log2x
x – 5 5
y
– 5
5
Graph on the same coordinate axes.
y = 2x
x y
0 1
1 2 –1 0.5
a 2. = xf x Let = y f x
x – 5 5
y
– 5
5
Graph on the same coordinate axes.
y = 2x
y = log2
x
x y
1 0
2 1 0.5 –1
2 = lb. ogf x x Let = y f x
Section 10.4Exercise #12
Chapter 10
Write the equation
a. 27 = 3x in logarithmic form.
b. log5 25 = x in exponential form.
Write the equation
a. 27 = 3x in logarithmic form.
log3 27 = x
b. log5 25 = x in exponential form.
5x = 25
Section 10.4Exercise #13
Chapter 10
Solve.
a. log4 x = –1
b. logx 16 = 2
x = 4–1
x =
14
x 2 = 16
x = ± 16
x = 4
Section 10.5
Common and Natural Logarithms
OBJECTIVES
A Find logarithms and their inverses base 10.
OBJECTIVES
B Find logarithms and their inverses base e.
OBJECTIVES
C Change the base of a logarithm.
OBJECTIVES
D Graph exponential and logarithmic functions base e.
OBJECTIVES
E Solve applications involving common and natural logarithms.
DEFINITIONNATURAL LOGARITHMIC FUNCTION
ƒ(x) = ln x, where x means loge x and x > 0
FORMULACHANGE-OF-BASE
logbM = logaMlogab
Section 10.5Exercise #18
Chapter 10
Use the change of base formula to fill in the blank.
log310 = _______
log310 =
log 10
log 3
=
10.4771
= 2.0959
log310 =
log 10
log 3
Use the result part a to find a numerical approximationfor log310.
Section 10.5Exercise #19
Chapter 10
x – 5 5
y
– 5
5
Graph.
12
xy = e
12a. =
xf x e
x y
0 1
1 1.6 –1 0.6 2 2.7
Section 10.5Exercise #20
Chapter 10
x – 5 5
y
– 5
5
ln + 1y = x
Graph.
= ln 1. + a f x x
x y
0 0
1 0.7 2 1.1
– 0.5 – 0.7
Graph.
= ln b. + 1g x x
x – 5 5
y
– 5
5 y = ln x + 1 x y
1 1
2 1.7 3 2.1
0.3 0.5
Section 10.5Exercise #21
Chapter 10
Solve.
x =
12
a. 52x + 1 = 25
52x + 1 = 52
2x + 1 = 2
2x = 1
If bm = bn , then m = n
x = 1
b. 3x + 1 = 92x – 1
x + 1 = 4x – 2
–3x = – 3
2 – 1 + 1 23 = 3xx
3x + 1 = 3 4x – 2
Solve.
Section 10.5Exercise #22
Chapter 10
Solve.
a. 3x = 2
x =
log 2
log 3
x =
0.30100.4771
x log 3 = log 2
x = 0.6309
Solve.
b. 50 = e0.20k
k = 19.56
ln 50 = 0.20k • ln e
0.39120 = 0.20k
ln 50 = ln e0.20k
Section 10.5Exercise #23
Chapter 10
Solve.
log + 2 + log – 7 1a. = x x
log + 2 – 7 = 1 x x
1 + 2 – 7 = 10x x
x2 – 5x – 14 = 10
x2 – 5x – 24 = 0
x – 8 = 0
x2 – 5x – 24 = 0
– 8 + 3 = 0x x
x + 3 = 0
x = 8 x = – 3
or
= – 3 causes log –3 + 2 = log –1–3, = 8
xx x
Solve.
log + 2 + log – 7 1a. = x x
Solve.
3 3 log + 5 – log – 1b. = 1x x
3 log + 5 ÷ – 1 = 1 x x
1 + 5 = 3
– 1
x
x
+ 5 = 3 – 1x x
x + 5 = 3x – 3
x + 5 = 3x – 3
8 = 2x
4 = x
x = 4
Solve.
b. log3 x + 5 – log3 x – 1 = 1
Section 10.5Exercise #24
Chapter 10
The compound amount with continuous
compounding is given by A= Pert , whereP is the principal, r is the interest rate, andt is the time in years. If the rate is 8%, findhow long it takes for the money to double--for A to equal 2P (ln 2 = 0.69315.)
A = 2P , r = 0.08 find t.
2P = Pe0.08t
A = Pert
2PP
= e0.08t
2 = e0.08t
2 = e0.08t
ln2 = 0.08 lnt e
0.69315 = 0.08t
8.66 = t
It takes 8.66 years to double the money.
Section 10.5Exercise #25
Chapter 10
A radioactive substance decays sothat the amount A percent at time
t (years) is A = A0e–0.5t . Find thehalf-life (time for half to decay) ofthis substance (ln 2 = 0.69315.)
A =
12
A0 , find t.
A = A0e–0.5t
12
A0 = A0e–0.5t
12
= e–0.5t
2–1 = e–0.5t
–1 ln 2 = – 0.5 lnt e
2–1 = e–0.5t
–0.69315 = – 0.5t
1.3863 = – 0.5t
1.386 years is the half-life of this substance.
Section 10.6
Exponential and Logarithmic Equations and Applications
OBJECTIVES
A Solve exponential equations.
OBJECTIVES
B Solve logarithmic equations.
OBJECTIVES
C Solve applications involving exponential or logarithmic equations.
DEFINITION
An equation in which the variable occurs in an exponent.
EXPONENTIAL EQUATION
DEFINITIONEQUIVALENCE PROPERTY
For any b > 0, b ≠ 1, bx = by
is equivalent to x = y.
DEFINITIONEQUIVALENCE PROPERTY FOR LOGARITHMS
logbM = logbN
is equivalent to
M = N
SOLVING LOGARITHMIC EQUATIONSPROCEDURE
1. Write equation: logbM = N2. Write equivalent exponential
equation. Solve.3. Check answer and discard
values for M ≤ 0.