Math Analysis - Chapter 5.1, 5.2, 5.3
Mathlete: ____________________
Date
Assigned
Section
Homework
(due the next day)
Mon
3/27
5.1 Inverse Functions
p.373: 9,13,15,17, 23-33odd, 37, 39, 43, 55, 59, 73,
76
Tue
3/28
5.2 Exponential Functions
p.387: 17, 21, 25, 33, 45-52, 61-67odd, 73
Wed
3/29
5.3 Logarithmic Expressions
p.403: 11, 13, 15-33 odd, 43-55 odd
Thu
3/30
5.3 Logarithmic Graphs
p.404: 57, 61, 63, 65, 75-78
Fri
3/31
Review 5.1, 5.2, 5.3
p.441-442: 1-21odd, 25-35odd, 41-47odd
No homework for Spring Break!
Mon
4/10
Review 5.1, 5.2, 5.3
(Finish assignment from 3/31)
p.441-442: 1-21 odd, 25-35 odd, 41-47 odd
Wed
4/12*
Quiz 5.1, 5.2, 5.3
✔
No School:
April 3-7: Spring Break
Tue April 11: No classes Seniors (PSAT/SAT for 9th-11th)
*Wed April 12: Late Start
Fri April 14: No School - Good Friday
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5.1 Inverse Functions
To verify that f and g are inverses of each other, we show that (f ° g )(x) = (g ° f )(x) .
Example 1: ​Verify that the given functions are inverses of each other.
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10. f (x) = x + 7; g(x) = x − 7
16. f (x) = x3 − 4; g(x) = √x + 4
18. f (x) = x2 − 7, x ≤ 0; g(x) = − √x + 7
Recall: ​A relation is a function if for each input x, there is exactly one output y. An informal way to say this is
“x-values cannot be repeated”. Graphically, a relation is a function if it passes the vertical line test.
Once you have a function f(x), you may be asked to determine whether the function has an inverse f​-1​(x).
Definition of a One-to-One Function
For a function f(x) to have an inverse function, f must be one-to-one. That is,
● algebraically: if f(a)=f(b), then a=b
● graphically: f must pass the ​horizontal line test
An informal way to tell if a function is one-to-one is that is must be strictly increasing or strictly decreasing.
Example 2: ​State whether each function given graphically is one-to-one.
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Example 3:​ State whether each function is one-to-one. Hint: draw a quick sketch.
30. f (x) = 34 x + 1
32. f (x) = − 3x2 + 1
34. f (x) = − 31 x3 − 5
Given a function f(x), to find the inverse f​-1​(x):
Algebraically
1. Switch the variables x and y.
2. Solve for y.
3. Replace y with the notation f​-1​(x).
Graphically
1. Switch the variables x and y.
2. Plot the points.
3. A function f(x) and its inverse f​-1​(x) are symmetric with respect to the line ​y=x​.
Example 4: ​Find the inverse of the given function. Then graph the given function and its inverse on the same
set of axes.
38. f (x) = 2x − 59
3
44. g (x) = − x2 + 3, x ≤ 0
Example 5: ​The graph of f is given. Sketch the graph of the inverse function f​-1​. State the domain and range of
f and f​-1​.
4
5.2 Exponential Functions
Definition of an Exponential Function
An exponential function is a function of the form
f (x) = a · bx
where a =/ 0, b > 0 and b =/ 1. The domain of an exponential function is all real numbers. The range will
vary depending on the values of a and b.
Example 1: ​Sketch the graph of each function.
f (x) = 2x
x
a = _____
b = _____
increasing or decreasing?
f (x) = 2x
-2
-1
0
y-intercept = _____
horizontal asymptote:
1
2
22. f (x) = 4(2)−x
x
a = _____
b = _____
increasing or decreasing?
f (x) = 4(2)−x
-2
-1
0
y-intercept = _____
1
horizontal asymptote:
2
5
34. f (x) = − 2(3)x + 1
x
a = _____
b = _____
increasing or decreasing?
f (x) = − 2(3)x + 1
-2
-1
0
y-intercept = _____
1
horizontal asymptote:
2
Compound Interest
Suppose an amount P is invested in an account that pays interest at rate r, and the interest is compounded n
times a year. Then after t years the amount in the account will be:
Continuously Compounded Interest
Suppose an amount P is invested in an account that pays interest at rate r, and the interest in compounded
continuously. Then after t years, the amount in the account will be:
Example 2: ​Suppose $2500 is invested in a savings account. Find the amount in the account after 4 years if
the interest rate is 5.5% compounded:
a. Monthly
b. Continuously
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5.3 Logarithmic Expressions
The inverse function of the exponential function is the logarithmic function.
Definition of a Logarithm
Let b>0 but not equal to 1. If y>0, then the ​logarithm of y with base b​ is
bx = y if and only if x = log b y
The number b is the base.
Question: ​What is the purpose of the logarithm?
Answer: ​It solves for the exponent!
Example 1: ​Complete the table by filling in equivalent exponential statements or logarithmic statements.
Logarithmic
statement
Exponential statement
Logarithmic
statement
40 = 1
log 3 9 = 2
log 5 √5 =
Exponential statement
1
2
10−1 =
1
10
log 2 41 = − 2
61/3 = √6
log a m = k , a > 0
ak = v , a > 0
3
Example 2: ​Evaluate the expression without using a calculator.
1
a. log 5 125
b. log 10 100
c. log a a4 , a > 0
d. 3log3 5
d. log 6 36
e. log 10 (− 1)
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Definition of a Common Logarithm
y = log x
Definition of a Natural Logarithm
y = ln x
Example 3: ​Evaluate the expression without using a calculator.
a. log 10, 000
b. lne1/2
c. eln a , a > 0
Change of Base Formula
log a x =
(in terms of base 10)
log a x =
(in terms of base e)
Use the change of base formula when you need base 10 or base e for your calculator.
Example 4:​ Use the change of base formula with the indicated logarithm to calculate the following:
a. log 6 15 using common log
b. log 7 0.3 using natural log
8
5.3 Logarithmic Graphs
You are familiar with graphing exponential functions. Today we will graph their inverses, the logarithmic
function.
The functions ​f(x)=b​x​ and ​g(x)=log​b​x​ are inverses of each other. The best way to understand a logarithmic
function is to start with an exponential function.
How do you graph the inverse? _____________________________
What is the symmetry line for a function and its inverse? _______________________
y = 2x
x
y = log 2 x
y
x
y
-2
-1
0
1
2
Domain: __________
Domain: __________
Range: __________
Range: __________
Asymptote: __________
Asymptote: __________
y = ex
y = lnx
x
y
x
y
-2
-1
0
1
2
Domain: __________
Domain: __________
Range: __________
Range: __________
Asymptote: __________
Asymptote: __________
When there is no horizontal shift, the domain a logarithmic function is: _________________________
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Properties of Logarithmic Functions
f (x) = log b x, b > 1
Domain:
Range:
Vertical asymptote:
Increasing on
(in terms of x)
Inverse function:
f (x) = log b x, 0 < b < 1
Domain:
Range:
Vertical asymptote:
Decreasing on
(in terms of x)
Inverse function:
Find the domain of each function. Graph the function and label all asymptotes.
58. f (x) = 4lnx
x
y = ex
x
f (x) = lnx
x
f (x) = 4lnx
-2
-1
0
1
2
Domain: __________
Range: __________
Asymptote: __________
10
62. f (x) = lnx + 2
x
y = ex
x
f (x) = lnx
x
f (x) = lnx + 2
f (x) = log 5 x
x
f (x) = log 5 (x − 2)
-2
-1
0
1
2
Domain: __________
Range: __________
Asymptote: __________
64. f (x) = log 5 (x − 2)
x
y = 5x
x
-2
-1
0
1
2
Domain: __________
Range: __________
Asymptote: __________
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