Secondary 2
Chapter 16
Secondary II
Chapter 16 – Other Functions and Inverses
2014/2015
Date
Section
Assignment
Concept
A: 3/20
B: 3/23
16.1 /16.2
- Worksheet 16.1 & 16.2
Piecewise functions
Step functions
A: 3/24
B: 3/25
16.3/16.4
- Worksheet 16.3 & 16.4
Inverses
A: 3/26
B: 3/27
3/30 - 4/3
A: 4/6
B: 4/7
A: 4/8
B: 4/9
Computer Lab
End of Term & Spring Break
Review
Chapter 16 TEST
Late and absent work will be due on the day of the review (absences must be excused). The review
assignment must be turned in on test day. All required work must be complete to get the curve on the test.
Remember, you are still required to take the test on the scheduled day even if you miss the review, so come
prepared. If you are absent on test day, you will be required to take the test in class the day you return. You
will not receive the curve on the test if you are absent on test day unless you take the test prior to your
absence
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Secondary 2
Chapter 16
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Secondary 2
Chapter 16
Chapter 16: Other Functions and Inverses
16.1-16.2 - Linear Piecewise and Step Functions
Example 1: Emily won $48 in a ping-pong tournament. For the first 5 days after her victory, she spent
$3 each day. Then she spent nothing for the next 5 days. After those 5 days, she spent $1.50 a day until
her prize winnings were gone.
a) Complete the table to show the amount of money Emily had remaining after each day.
Time Since
Tournament
(days)
Prize
Money
Remaining
(dollars)
Emily’s Spending of Ping-Pong Prize Money
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
b) Plot the points on a graph. Make sure you label you axes with the appropriate units.
c) Determine when Emily will spend all her prize money.
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Secondary 2
Chapter 16
The graph that you created represents a linear piecewise function. Recall that a linear piecewise
function is a function that can be represented by more than one function, each of which corresponds to
a part of the domain.
d) What is the domain of this problem?
e) How many pieces make up this function?
f)
What is the domain of each piece?
g) Determine the equation that represents each piece of the function for each given time period.
From 0 to 5 days
From more than 5 days to
10 days
From more than 10 days to
32 days
To write piecewise functions you must write the equation and domain for each piece of the function
h) Write the piecewise function for the problem situation
𝑓(𝑥) =
{
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Secondary 2
Chapter 16
Example 2: Write a piecewise function to represent the data shown
X
0
1
2
3
4
5
6
7
8
9
F(x)
60
55
50
45
45
45
45
43
41
39
Example 3: Graph the following linear piecewise function , f(x), below. Then complete the function
notation problems.
Hint: The domain to the right of each piece tells you for what x-values the equation is used for, the
domain also specifies if open or closed dots should be used at the end of each piece.
𝑓(𝑥) = {−2𝑥 + 1 𝑥 ≤ 2
5𝑥 − 4 𝑥 > 2
a) f(3) =
b) f(2) =
c) f(-4)=
1. d) f(2)=
e) For what x value is 𝑓(𝑥) equal to 5?
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Secondary 2
Chapter 16
Step Functions
Example 4: In 2006, the rates for a taxi ride in Macon, Georgia, was $1.20 for the first mile or part of a
mile, and $1.20 for each additional mile or part of a mile.
a) Determine a piecewise function, g(x), for the cost of a taxi ride up to 5 miles.
b) What is the slope of each interval?
c) Graph g(x) for x<5 miles
d) Describe the graph of the function as either increasing or decreasing.
You have just graphed a step function. A step function is a piecewise function whose pieces are disjoint
constant functions.
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Secondary 2
Chapter 16
Example 5: Postal Rates (First class mail)
A letter weighing up to one ounce will cost $0.45 to mail
A letter weighing more than one ounce and up to two ounces will cost $0.65 to mail
A letter weighing more than two ounces and up to three ounces will cost $0.85 to mail
Write a step function f(x) to describe this situation and graph it on calculator
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Secondary 2
Chapter 16
Additional Notes
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Secondary 2
Chapter 16
16.3-16.4 - Inverses of Linear and Non-Linear Functions
The inverse of a function will “undo” the algebra of the original function. An inverse is “opposite
algebraically”. The symbol for the inverse of a function is: 𝒇−𝟏 (𝒙)
a. What is the opposite of adding 5 to x?
b.
What is the opposite of squaring x?
Example 1: Miguel is planning a trip to Turkey. Before he leaves, he wants to exchange his money to
the official currency of Turkey. The exchange rate at the time of his trip is 2 Iira per 1 U.S. dollar.
a) What are the independent and dependent quantities?
b) Complete the table of values to show the conversion
U.S. Currency (dollars)
Turkish Currency (Iira)
100
250
400
650
1000
c) Write an equation to represent the number of Iira in terms of the number of U.S. dollars. Let r
represent the number of Iira, and let d represent the number of U.S. dollars.
d) What is the inverse of this problem?
Complete the table for the inverse of the problem
Turkish Currency (Iira)
U.S. Currency (dollars)
200
500
800
1300
2000
e) Write an equation for the inverse?
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Secondary 2
Chapter 16
Steps for finding the inverse of a function:
1. Replace the notation 𝑓(𝑥) with another variable, generally y
2. Switch the x and the y variables in the equation. (All x variables become y and all y variables
become x.)
3. Solve for y and if y is a function, replace the y variable with 𝑓 −1 (𝑥).
Graphically inverses are symmetric about y = x which means the ordered pairs just flip.
What does that mean for domain and range values??
Example 2: Find the inverse and check graphically (look at graph and table)
a) 𝑓(𝑥) = 8𝑥
b)
𝑓(𝑥) = 3𝑥 + 6
1
2
c) 𝑓(𝑥) = 𝑥 + 5
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Secondary 2
Chapter 16
Now how do we determine inverses of non-linear functions? A function needs to be one-to-one in order
to have an inverse. This means it must pass the horizontal and vertical line tests, or there is no
repeating x or y values. If an equation is not one-to-one you can restrict the domain to make the
equation one-to-one therefore it will have an inverse.
Example 3: For the given function complete the table and determine if it is one-to-one. Then check
graphically
a) f(x)= 3x-6
x
-2
-1
0
1
2
f(x)
x
𝑓 −1 (𝑥)
-2
-1
0
1
2
f(x)
x
𝑓 −1 (𝑥)
-2
-1
0
1
2
b) 𝑓(𝑥) = 2𝑥 2 + 2
x
-2
-1
0
1
2
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Secondary 2
Chapter 16
Example 4: Graph the function and restrict the domain so that the function is one-to-one. Then find the
inverse and list domain and range for f(x) and 𝑓 −1 (𝑥).
a.
𝑓(𝑥) = −4𝑥 2 − 2
𝑓(𝑥) = −4𝑥 2 − 2
𝑓 −1 (𝑥) =______________
Domain_____________________
Domain______________________
Range_______________________
Range_______________________
b. 𝑓(𝑥) = 𝑥 2 + 3
𝑓(𝑥) = 𝑥 2 + 3
𝑓 −1 (𝑥) =______________
Domain_____________________
Domain______________________
Range_______________________
Range_______________________
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Secondary 2
Chapter 16
Additional Notes
13