Unit 11- Piecewise Functions │ Page 1
Unit 11 Day 1
Piecewise Introduction
Emma wanted to show how math relates to just about everything she does. A graph that represents the amount of
water in her bathtub is shown. Time is measured on the x-axis and the amount of water is measured on the y-axis.
Describe each of the 3 intervals IN CONTEXT:
First:
Second:
Third:
Did the bathtub fill or drain faster? How do you know?
Suppose Emma fills the bathtub at a rate of 6 gallons per minute for 2 minutes. She lets the bathtub sit for 10
minutes. Then she lets the bathtub drain the water out of at a rate of 3 gallons per minute for 4 minutes. Use this
information to write the equation for each section of the graph.
First Section Equation:
Second Section Equation:
Third Section Equation:
For what times?
For what times?
For what times?
A Piecewise function is a function from “pieces” or “sections” of other functions typically written like this: Write the
piecewise equation for the amount of water in the bathtub.
𝐴(𝑡) =
{
𝑓𝑖𝑟𝑠𝑡 𝑠𝑒𝑐𝑡𝑖𝑜𝑛
𝑠𝑒𝑐𝑜𝑛𝑑 𝑠𝑒𝑐𝑡𝑖𝑜𝑛
𝑡ℎ𝑖𝑟𝑑 𝑠𝑒𝑐𝑡𝑖𝑜𝑛
𝑑𝑜𝑚𝑎𝑖𝑛
𝑑𝑜𝑚𝑎𝑖𝑛
𝑑𝑜𝑚𝑎𝑖𝑛
Write the equation for the piecewise function:
(To help, separate the graph into sections)
𝑓(𝑥) =
{
𝐴(𝑡) =
{
Unit 11- Piecewise Functions │ Page 2
Graph the given piecewise function:
2𝑥,
𝑥≥1
𝑓(𝑥) = {𝑥 − 1, −2 < 𝑥 < 1
−1,
𝑥 ≤ −2
What is 𝑓(2) using the graph and the function?
What is 𝑓(−4) using the graph and the function?
What is 𝑓(0) using the graph and the function?
What is 𝑓(−3) using the graph and the function?
Unit 11 Assignment 1
Directions: Graph the function and fill in the blank below.
3)
𝑓(2) =_______
4) 𝑘(𝑥) = 2|𝑥 + 4| − 3
When 𝑔(𝑥) = −1, 𝑥 =______
1
5) 𝑚(𝑥) = 𝑥 − 3
ℎ(−1) =_____
6) (𝑥 + 3)2 + (𝑦 − 1)2 = 4
If 𝑘(𝑥) = 3, then 𝑥 =_____
𝑚(8) =_____
When 𝑥 = −2, 𝑦 =_____
7)
𝑓(𝑥) = 𝑥 + 1
ℎ(𝑥) = 2𝑥 + 1
2) 𝑔(𝑥) = (𝑥 + 1)(𝑥 − 3)
1)
𝑏(𝑥) = −(𝑥 + 4)2 + 2
4
8) 𝑓(𝑥) = {1
𝑓(−4) =
2
𝑥 − 4, 𝑖𝑓 𝑥 > 3
𝑥 + 1,
𝑖𝑓 𝑥 ≤ 3
𝑓(3) =
𝑓(2) =
𝑓(5) =
When 𝑓(𝑥) = 4, 𝑥 =
>>>>>>>>CONTINUED ON NEXT PAGE >>>>>>>
2𝑥 + 1, 𝑖𝑓 𝑥 > 0
9) 𝑓(𝑥) = {
−𝑥 + 1, 𝑖𝑓 𝑥 ≤ 0
Find 𝑥 where 𝑓(𝑥) = 5.
10) 𝑔(𝑥) =
1
− 𝑥
{ 2
Unit 11- Piecewise Functions │ Page 3
𝑥 2 , 𝑖𝑓 0 < 𝑥 ≤ 2
11) ℎ(𝑥) = { 1, 𝑖𝑓 𝑥 ≤ 0
−2, 𝑖𝑓 𝑥 > 2
− 1, 𝑖𝑓 𝑥 ≤ 2
2𝑥 − 5,
𝑖𝑓 𝑥 > 2
𝑔(3) =
ℎ(0) =
Directions: Use your knowledge from previous units to solve the following problems.
12) Solve for x. 𝑥 2 + 5𝑥 + 4 = 0
14) Find all zeros.
4𝑥 2 = 36
16) A 4 m by 6 m rug covers
half of the floor area of a
room and leaves a
uniform strip of bare
floor around the edges.
What are the dimensions
of the room?
18) Which symbol (<, >, =) belongs in the space if 𝑥 =
−2?
𝑓(𝑥) = 𝑥 2 + 18___ 𝑔(𝑥) = 2(𝑥 − 1)2
20) Midge wants to know if GPA and gender are
independent. She finds that in her school 48% of all
students have a GPA of 3.0 or better. What else does
she need to do to test for independence?
22) Jim is experimenting with a new drawing program on
his computer. He created quadrilateral 𝑇𝐸𝐴𝑀 with
the coordinates 𝑇(−2,3), 𝐸(−5, −4), 𝐴(2, −1), and
𝑀(5,6). Jim believes that he has created a rhombus
but NOT a square. Prove that Jim is correct.
(remember “looks like” is not a proof!)
13) Solve for x. 3𝑥 2 = −𝑥
15) Simplify: 3 − 2(5 + 2)
17) For the problem:
In southern California,
- Make a drawing
there is a six mile section of
- Write an equation
Interstate 5 that increases
- Solve (do not
2500 feet in elevation.
forget to include
What is the angle of
units of measure)
elevation?
24) You have been given three corresponding parts of
two congruent triangles. Using the parts, sketch a
picture of the triangles, then write a congruence
statement and state the Theorem that proves the
triangles are similar.
̅ ≅ ̅̅̅̅
 𝐽𝐿
𝐹𝐸 , ̅̅̅̅
𝐺𝐸 ≅ ̅̅̅̅
𝐻𝐿, ∠𝐸 ≅ ∠𝐿
̅̅̅̅
̅̅̅̅
 𝐺𝐹 ≅ 𝐽𝑀, ∠𝐹 ≅ ∠𝑀, ∠𝐽 ≅ ∠𝐺
26) Identify the pattern in
𝑓(𝑥)
the tables as linear,
Pattern:
X
Y
quadratic, or
-3 -23
exponential. If
-2 -17
possible, write an
Equation:
-1 -11
equation. Look at the
0
-5
patterns in
1
1
2
7
25) Given CD = 2.5 cm,
𝑚∠𝐵 = 40°
Find the length of BD.
19) List the sample space of drawing two letters, without
replacement, from the word “PILLOW.”
21) Find the vertex for the given function. State whether
it’s a maximum or minimum.
3
𝑦 = − 𝑥 2 + 9𝑥 + 25
2
23) Given circle K and the
marked angle measure,
find the area of the small
sector as well as its arc
length. Write your
answers in terms of 𝜋.
𝑔(𝑥)
Pattern:
Equation:
ℎ(𝑥)
X
-3
-2
-1
0
1
2
Y
4
0
-2
-2
0
4
Pattern:
Equation:
X
-3
-2
-1
0
1
2
Y
-15
-10
-5
0
5
10
Unit 11- Piecewise Functions │ Page 4
U11 Day 2
Linear Piecewise Story Problems and Step Functions
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 days, she spent $1.50 a day until her prize winning were gone.
Create a table to display
Use your table to create a graph that models the data. Be
a. When will Emily
Emily’s spending habits.
sure to label the axis!
spend all of her
prize money?
Time Since
Tournament
(days)
Prize Money
Remaining
(dollars)
b. When will Emily
have $40?
0
1
2
3
4
c. What is the
Domain of the
graph?
5
6
7
8
d. What is the Range
of the graph?
9
10
11
e. Solve 𝑓(10)
12
13
14
What does this mean
in context?
Write an equation for each section of the graph.
f. 0 to 5 days
g. 5 to 10 days
h. More than 10 days
i.
Write the piecewise function:
𝑚(𝑑) =
{
Step Function
The Step Functions is a function that takes any number and rounds it DOWN to the nearest INTEGER. It looks like this.
𝑓(𝑥) = ⌊𝑥⌋
For example:
19
⌊1.5⌋
⌊4⌋
⌊3.8454⌋
⌊−2.1⌋
⌊𝜋⌋
⌊ ⌋
8
Unit 11- Piecewise Functions │ Page 5
Unit 11 Assignment 2
Directions: Read the story, then answer all parts to the question.
1)
Jin fills up a 510-gallon pool in the backyard for her children. She fills it with the garden hose at a rate of 17 gallons per
minute. After it is filled, she lets it sit for 30 minutes in order to let the water temperature rise. The kids then get in and have
fun for an hour. The pool loses about ½ gallon of water each minute due to their splashing and playing. At the end of the
hour, they tear the pool while getting out, which causes a leak. The pool then begins to lose water at a rate of 2 gallons per
minute.
a. Complete the table to show the
b. Create a graph to model the problem situation.
d. Identify the x- and yamount of water in the pool
Extend your graph to include when the pool will
intercept then explain what
after each minute.
be empty. Label the graph.
they mean in terms of the
problem situation.
Time
Amount of Water
(minutes)
(gallons)
0
5
20
30
45
60
80
e.
100
120
150
200
c.
Determine when the pool
will have 470 gallons of
water in it. Identify the
piece(s) of function you
used. Explain your reasoning.
Write a piecewise function that models this
problem situation. Explain your reasoning for
each piece of the function.
𝑓(𝑥) =
{
2)
The graph shows the distance, 𝑑, of two cyclists from their starting point over time, 𝑡, during an all-day cycling event.
b. In context, describe what is
e. If the cyclists are always moving
happening from 3 to 5 hours.
away from, or towards the finish
line, how many total miles have the
cyclists traveled after 8 hours?
c. In context, describe what is
happening from 5 to 8 hours.
d.
𝑓(𝑥) =
a.
In context, describe what is
happening from 0 to 3 hours.
f.
When will a cyclist be 6 miles from
the starting point?
g.
Write the previous statement in
function notation.
Complete the piecewise function to
represent the graph.
𝑓(___) =_____
{
Directions: Evaluate the following. Take care in noticing if it is a step function or an absolute value function!
3)
⌊5.99⌋
4) 2|−8 + 4|
5) ⌊−3.1⌋
6) |4.44|
Unit 11- Piecewise Functions │ Page 6
>>>>>CONTINUED ON NEXT PAGE >>>>>>>
Directions: Given the piecewise function, evaluate it for the given values.
7)
3𝑥 + 2, 𝑖𝑓 𝑥 ≥ 2
(𝑥 − 2)2 , 𝑖𝑓 − 5 ≤ 𝑥 < 2
𝑓(𝑥) = {
1
𝑥 + 5, 𝑖𝑓 𝑥 < −5
2
𝑓(5) =_______
𝑓(−2) =_____
8)
−2𝑥 + 4,
𝑥≥1
𝑓(𝑥) = { 𝑥 − 1, −2 < 𝑥 < 1
−1,
𝑥≤2
𝑓(0) =_______
𝑓(6) =_____
9)
|2𝑥| − 1,
𝑥≥0
1
𝑓(𝑥) = { 3 𝑥 + 8, −3 < 𝑥 < 0
𝑥 ≤ −3
3𝑥 2 ,
𝑓(−1) =_______
𝑓(−3) =_____
Directions: Use your knowledge from previous lessons to solve the problems below.
10) A rectangular field will be fenced on all four sides.
There will also be a line of fence across the field,
parallel to the shorter side. If 900 m of fencing is
available, what dimensions of the field will produce
the maximum area? What is the maximum area?
12) If 𝑓(𝑥) is a linear
function, 𝑔(𝑥) is not.
- Label the two
functions.
- Find 𝑥 when
𝑓(𝑥) = 𝑔(𝑥)
- For what values is
𝑓(𝑥) > 𝑔(𝑥)?
- 𝑔(3) =____
11) Use the function to find the missing values. Verify
with a graph.
𝑓(𝑥) = 𝑥 2 − 6𝑥 + 9
𝑓(0) =____
𝑓(−3) =_____
𝑓(𝑥) = 0, 𝑥 =____
𝑓(𝑥) = 16, 𝑥 =_____
13) Find the probability of achieving success with each of
the events below. (answer as simplified fractions)
 Rolling an even number on a standard die.
 Drawing a black ace from a deck of cards.
 Rolling a die twice in a row and getting two
threes.
14) A hot air balloon is 100 feet straight above where it
is planning to land. Sara is driving to meet the
balloon when it lands. If the angle of elevation from
her to the balloon is 35°, how far away is Sara from
where the balloon will land?
15) Use the information below to create a two-way table.
16) Solve the
 |3𝑑| + 6 = 15
 Data was collected at the movie theater last fall. Not about
 −9|𝑚| = −63
absolute
movies, but about clothes.
 −|𝑚 + 3| = −13
value
 6525 people were observed.
functions to
 3123 had on shorts, the rest had on pants.
the right.

45% of the shorts were denim.
 Of those wearing pants, 88% were denim.
17) Oscar's dog house is shaped like a tent. The slanted sides are
both 5 feet long and the bottom of the house is 6 feet across.
What is the height of his dog house, in feet, at its tallest point?
18) Solve each system of equations
𝑓(𝑥) = 24𝑥 − 𝑥 2
7𝑥 + 𝑦 = 12
{
𝑦 = 5𝑥 − 36
𝑔(𝑥) = 8𝑥 − 48
2
20) Simplify: 1 + (4 − 2)(−7 + 9)3 − 1
19) Simplify: 2(5) − 4
21) The Sears Tower in Chicago is 1730 feet tall. If a penny were let go from the top of the tower, the position above
the ground 𝑠(𝑡) of the penny at any give time 𝑡 would be 𝑠(𝑡) = −16𝑡 2 + 1730.
Fill in the missing positions in both charts.
 How high above the ground is the penny after 7
𝑠(𝑡) : distance
total distance
seconds have passed?
from ground
fallen
𝑡
𝑡
0
1730
0
0
 How far has it fallen when 7 seconds have passed?
1
1714
1
16
 Has the penny hit the ground at 10 seconds?
2
2
3
3
 Put 𝑠(11) = −206 into context for the problem.
4
4
5
5
6
6
7
7
8
8
9
9
{
Unit 11- Piecewise Functions │ Page 7
10
10
U11 Day 3
Graphing Step Functions and Quadratic Piecewise
Graphing Step Functions (at least 3 steps)
The Step Function is a function that takes any number and rounds it DOWN
to the nearest INTEGER. It looks like this.
𝑓(𝑥) = ⌊𝑥⌋
To graph it, let’s try some values for x to get an idea.
𝑥
0
.5
1
1.5
-.5
-1
-1.5
𝑓(𝑥)
1. 𝑓(𝑥) = 2⌊𝑥⌋
𝑥
𝑓(𝑥)
Graphing Other Step Functions
2. 𝑔(𝑥) = ⌊𝑥 − 2⌋
Step Functions as Story Problems.
3. ℎ(𝑥) = ⌊𝑥⌋ − 2
Unit 11- Piecewise Functions │ Page 8
In 2006, a taxi ride in Macon, Georgia, was $1.20 for the
first mile or part of a mile, and then $1.20 for each
additional mile or part of mile.
1.20, 𝑓𝑜𝑟 0 ≤ 𝑥 < 1
𝑔(𝑥)
{
Minimum Bill Due: Write a piecewise function that
models the situation below.
A local store offers store credit with the following
stipulations in each billing cycle.
- If the bill is less than $25, the customer must pay
the full amount due.
- If the bill at least $25, but less than $50 the
cutomer pays $25.
- If the bill is $50 or more, the cutomer must pay
half the amount due.
𝑓(𝑥) =
{
Unit 11 Assignment 3
Directions: Read the story then answer all parts to the question.
1) A kids bounce house charges $8 for the first hour and $2 for each additional hour of playtime. Create a graph and
write a function that represents the charges for up to 5 hours of playtime (be sure to label your axis!).
Function: 𝑓(𝑥) =
{
2)
A department store offers store credit, but has the following rules for billing each month. For a bill less than $15 the
customer pays the entire amount due. For a bill of at least $15 but less than $50, the minimum due is $15. For a bill of at
least $50 but less than $100, the minimum the customer pays is $20. For a bill of $100 or more, the minimum due is 25% of
the entire bill.
a. Complete the piecewise
b. Graph (label axis!)
c. Is this piecewise function a
function 𝑓(𝑥) for the
step function? Why or why
minimum amount due for the
not?
amount of bill 𝑥.
𝑓(𝑥) =
d.
{
A customer comes in the store
to pay the minimum due on
his bill of $100. The customer
thinks he owes $20, but the
cashier tells him he owes $25.
Who is correct? Explain your
reasoning.
Directions: Evaluate the following using these functions: 𝒇(𝒙) = ⌊𝒙⌋, 𝒈(𝒙) = ⌊𝒙⌋+. 𝟓, 𝒉(𝒙) = 𝟐⌊𝒙⌋
3) 𝑓(2)
1
2
4) 𝑓 ( )
5) 𝑔(3.7)
6) ℎ(1)
Unit 11- Piecewise Functions │ Page 9
7) ℎ(−5.7)
8) 𝑓(−2.3)
Directions: Use your knowledge from previous lessons to solve the problems below.
9) Seth is on sophomore committee and they are selling
tickets to a school dance for $4 per person and the
projected attendance is 300 people. For every $1
increase in ticket price, 𝑡, the dance committee
projects that attendance will decrease by 5. Seth
wants to see if they can make more money, 𝑅(𝑡), so
he creates a function to show this pattern:
𝑅(𝑡) = (300 − 5𝑡)(4 + 𝑡)
Use the function to find the ticket price and maximum
revenue for the sophomore committee.
10) Write the
function to the
right as an
absolute value
function.
12) For the function ℎ(𝑥) = 2𝑥 + 𝑥 2 , build a table of
values using 𝑥 = {−2, −1,0,1,2}. Write all values as
fractions.
13) A survey was taken to determine how many students
own dogs and cats as pets. When a student from the
survey is randomly chosen, 𝑃(both 𝐶 and 𝐷) =
1⁄ , 𝑃(𝐷) = 1⁄ , and 𝑃(𝐶) = 1⁄ .
4
12
3
 What is the probability a student owns a dog or
cat?
 What is the probability that a student chosen at
random who owns a dog, also owns a cat?
 Are “owning a dog” and “owning a cat”
independent events? Justify with calculations.
16) In the parallelogram below, solve for 𝑥 and 𝑦.
14) Explain why |𝑚| = −3 has no solution.
15) Find 𝑥 in the figures below.
11) Write the
function to the
right as a
piecewise
function.
>>>>> CONTINUED ON NEXT PAGE >>>>>
Directions: match the function on the left with the equivalent function on the right.
17) 𝑓(𝑥) = −2𝑥 + 5
18)
19) I put $7000 in a savings account that pays 3%
interest compounded annually. I plan to leave it in
the bank for 20 years.
20) The area of the triangles below.
a.
b.
Unit 11- Piecewise Functions │ Page 10
c. 𝑦 + 𝑥 = 0
d. 𝑦 = (𝑥 − 1)(𝑥 + 3)
e. 𝐴 = 7000(1.03)20
f.
21) 𝑓(0) = 5; 𝑓(𝑛) = 2 ∙ 𝑓(𝑛 − 1)
22) 𝑓(0) = 5; 𝑓(𝑛) = 𝑓(𝑛 − 1) − 2
23)
𝑥
𝑓(𝑥)
g. 𝑓(𝑥) = 5(2)𝑥
-7.75
7.75
− 1⁄4
1⁄
4
1⁄
2
− 1⁄2
11.6
-11.6
𝑓(1) = 2;
𝑓(𝑛 + 1) = 𝑓(𝑛) + 2𝑛
Unit 11- Piecewise Functions │ Page 11
U11 Day 4
More Piecewise and Function Notation
Graphing Linear and Quadratic Piecewise Functions
Write the piecewise function of the graph:
To help with this separate the piecewise into all the sections. Then write down their respective domains.
𝑓(𝑥) =
{
Evaluate the following:
4. 𝑓(−10)
5. 𝑓(−5)
6. 𝑓(0)
7. 𝑓(3)
8. 𝑓(7)
Graph the following Piecewise Function:
𝑚(𝑥) =
{
−4
5
𝑥+2
4
⌊𝑥⌋
3
− |𝑥 + 1| + 12
2
− ∞ < 𝑥 ≤ −4
−4<𝑥 <2
2≤𝑥<6
𝑥≥6
Given the following piecewise evaluate the following:
−30
− ∞ < 𝑥 ≤ −4
25𝑥 + 35
−4<𝑥 <2
𝑓(𝑥) = {
⌊2𝑥⌋
2 ≤ 𝑥 < 14
2
𝑥 − 14
𝑥 ≥ 14
To help evaluate problems 6 through 10 determine which function (1st, 2nd , 3rd or 4th ) section you will need to use to
evaluate.
1. 𝑓(−10)
2. 𝑓(0)
3. 𝑓(11.77)
4. 𝑓(−4)
5. 𝑓(20)
Unit 11- Piecewise Functions │ Page 12
Unit 11 Assignment 4
Directions: Read the story then answer all parts to the question.
1)
To encourage quality and minimize defects, a manufacturer pays his employees a bonus based on the value of defective
merchandise produced. The fewer defective merchandise produced, the greater the employee’s bonus.
Employee bonus calculated by:
a. Graph (label axis!)
b. Complete the piecewise function
 $50 bonus for defective
𝑓(𝑥) for the employee’s bonus,
merchandise more than $0 and
given the amount of defective
up to and including $100,
merchandise 𝑥.
 $30 for more than $100 and up to
and including $200 of defective
merchandise,
𝑓(𝑥) =
 $10 for more than $200 and up to
and including $300 of defective
{
merchandise, and

$0 for more than $300 of
defective merchandise
Directions: Use your knowledge of piecewise functions to complete the tasks below.
2)
Graph the following piecewise function:
−2(𝑥 + 4)2 + 3
− ∞ < 𝑥 ≤ −2
5
𝑔(𝑥) = {
− 𝑥+3
−2 <𝑥 < ∞
2
3) Write the function to the following piecewise function:
𝑓(𝑥) =
{
4)
Graph the following piecewise function:
3𝑥 + 2, 𝑖𝑓 𝑥 ≥ −1
(𝑥 − 2)2 , 𝑖𝑓 − 4 ≤ 𝑥 < −1
𝑓(𝑥) = {
1
𝑥 + 5, 𝑖𝑓 𝑥 < −4
2
5) Write the function for the following piecewise graph:
𝑓(𝑥) =
{
6) Evaluate the piecewise function in #2 for the given values.
𝑔(5) =_____
𝑔(0) =_____
𝑔(−6) =_____
𝑔(3.5) =_____
7) Evaluate the piecewise function in #4 for the given values.
𝑓(5) =_____
𝑓(0) =_____
𝑓(−6) =_____
𝑓(3.5) =_____
>>>>> CONTINUED ON NEXT PAGE >>>>>
Unit 11- Piecewise Functions │ Page 13
Directions: Use your knowledge from previous lessons to solve the problems below.
8) A water balloon is catapulted into the air so that its
9) Find 𝑥, 𝑟, 𝑠, and 𝑞 in the figures below.
2
height ℎ, in meters, after 𝑡 seconds is ℎ = −4.9𝑡 +
27𝑡 + 2.4. Answer the following:
- How high is the balloon after 1 second?
- For how long is the balloon more than 30 m high?
- What’s the maximum height of the balloon?
- When will the balloon burst as it hits the ground?
10) Help Jason finish the last problem on his homework.
He is supposed to use the table and graph to write all
three forms of the quadratic function.
𝑥
-5
-4
-3
-2
-1
0
1
2
3
𝑦
10
5
2
1
2
5
10
17
26
12) Solve the quadratic function using an appropriate
method. (square root, factoring, quadratic formula)
 𝑚2 + 15𝑚 + 56 = 0
 2𝑥 2 + 32 =0
 12𝑥 + 14 = 2𝑥 2
 (𝑥 + 4)2 − 1 = 15
14) Given the similar triangles below, solve for all missing
sides.
11) State whether each statement is true or false. If it is
false, explain why.
a. All squares are also rectangles.
b. A rhombus is always a square.
c. If a figure is a trapezoid, then it is also a
parallelogram.
d. The diagonals of a rectangle bisect the angles.
e. A parallelogram can have 3 obtuse angles.
f. The figure made by two pair of intersecting
parallel lines is always a parallelogram.
g. All of the angles in a parallelogram can be
congruent.
h. A diagonal always divides a quadrilateral into two
congruent triangles.
13) Abby is standing at the top of a very tall skyscraper
and looking through a telescope at the scenery all
around her. The angle of decline on the telescope
says 35° and Abby knows she is 30 floors up and each
floor is 15 feet tall. How far from the base of the
building is the object that Abby is looking at?
15)
Directions: Use the table of information to answer the questions that follow.
Data collected from 200 individuals concerning whether or not
to extend the length of the school year.
For
Against
No Opinion
Total
Youth (5 to 19)
7
35
12
Adults (20 to 55)
30
27
20
Seniors (55+)
25
16
28
Total
200
16) Complete the “total” column and row.
17) Given the condition that a person is an adult, what is the
probability that they are in favor of extending the school
year? P(For│Adult)=
18) Given the condition that a person is against extending the
school year, what is the probability they are a senior?
P(Senior│Against)=
19) What is the probability that a person has no opinion given
they are a youth? P(No Opinion│Youth)
Unit 11- Piecewise Functions │ Page 14
Unit 11 Day 5
Operations with Functions
Function Notation
Using the functions 𝑓(𝑥) and 𝑔(𝑥) to the right, fill in the following tables:
𝑥
-5
-3
-1
0
1
3
5
𝑓(𝑥)
𝑥
-5
-3
-1
0
1
3
5
𝑔(𝑥)
Using the tables and graph from above simplify these:
1. Fill in the table for 𝑚(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)
𝑥
Work
2. Fill in the table for 𝑛(𝑥) = 𝑓(𝑥) ∗ 𝑔(𝑥)
𝑥
𝑚(𝑥)
𝑛(𝑥)
-5
-3
-5
-3
-1
-1
0
1
0
1
3
3
5
5
3. Graph 𝑚(𝑥).
Work
4. Graph 𝑛(𝑥).
Given the functions 𝑓(𝑥) and 𝑔(𝑥) evaluate the following: 𝑓(𝑥) = −2|𝑥 + 1| + 5
𝑔(𝑥) = −𝑥 2 + 4
6. 𝑓(1) + 𝑔(3)
7. 2𝑓(−3) + 𝑔(1)
8. 𝑓(−5) − 𝑔(−3)
9. 2𝑓(2) + 3𝑔(0)
10. 𝑓(−1) ∗ 𝑔(5)
11. 2𝑔(1) − 𝑓(3) + 5
Unit 11- Piecewise Functions │ Page 15
Unit 11 Assignment 5
Directions: Read the story then answer all parts to the question.
1) A department store has an online site that customers can order from. The shipping rates are calculated as follows:
Rates:
a. Write a piecewise function 𝑓(𝑥) c. Graph the function. Be sure to
for the shipping cost for weight
label the axis!
 for a package that weighs no
of the package 𝑥.
more 10 pounds the cost is $5,
 for a package that weighs more
than 10 pounds but no more
than 20 pounds the cost is $10,
b. How much does a package
 for a package that weighs more
weigh if the shipping cost is
than 20 pounds but no more
$27.30?
than 30 pounds the cost is $15,
 for a package that weighs more
than 30 pounds, the cost is 70%
of the weigh
2) Isaac lives 3 miles away from his school. School ended at 3 pm and Isaac began his walk home with his friend Tate
who lives 1 mile away from the school, in the direction of Isaac’s house. Isaac stayed at Tate’s house for a while
and then started home. On the way he stopped at the library. Then he hurried home. The graph shows Isaac’s
distance from home after school.
a. Label each part of the graph with the corresponding
part of Isaac’s journey from the paragraph above.
b. How much time passed between school ending and
Isaac’s arrival home?
c. How long did Isaac stay at Tate’s house?
d. How far is the library from Isaac’s house?
e. Where was Isaac 3 hours after school ended?
f.
Use function notation to write a mathematical
expression that means the same thing as question e.
g. When was Isaac walking the fastest?
Directions: Given 𝒇(𝒙) = −𝟐𝒙 + 𝟕, 𝒈(𝒙) = 𝟐(𝒙 − 𝟑)𝟐 , and 𝒉(𝒙) = |𝒙| + 𝟒, perform the following operations.
1
2
3) 𝑓(1) + 𝑔(3)
4) 2𝑓(−3) + ℎ(−2)
5) 𝑓(−5) − ℎ(3)
6) 3𝑔(0) + ℎ(2)
7) 𝑓(1) ∙ 𝑔(2)
8) (𝑔(2))!
9) 𝑓(𝑔(2))
10) 𝑓 (ℎ(𝑔(3)))
Directions: Given the graph of m(x) and n(x), create a table for k(x) then graph it on the same coordinate plane as m(x) and n(x).
11)
12)
𝑥
-2
-1
0
1
2
𝑘(𝑥) = 𝑚(𝑥) + 𝑛(𝑥)
𝑚(𝑥) 𝑛(𝑥) 𝑘(𝑥)
𝑥
-2
-1
0
1
2
𝑘(𝑥) = 𝑚(𝑥) ∙ 𝑛(𝑥)
𝑚(𝑥) 𝑛(𝑥) 𝑘(𝑥)
>>>>continued on next page>>>>>
Unit 11- Piecewise Functions │ Page 16
Directions: Use your knowledge of piecewise functions to complete the tasks below.
13) Graph the following piecewise function:
4
− ∞ < 𝑥 ≤ −2
1 2
𝑔(𝑥) = { − 𝑥 + 2 − 2 < 𝑥 < 1
2
3𝑥 − 4
1≤𝑥<∞
14) Write the function to the following piecewise function:
2𝑥
𝑓(𝑥) =
{
Directions: Use your knowledge from previous lessons to solve the problems below.
15) The sum of the squares of two consecutive even
integers is 452. Find the two integers by solving the
function: 𝑥 2 + (𝑥 + 2)2 = 452
17) Roy and Travis are truck driving brothers. Roy leaves
Pecos, Texas headed for Dallas 450 miles away. Roy
leaves one hour before his brother and averages 55
miles per hour. Travis leaves Dallas during rush hour
so he only averages 45 miles per hour.
a. Label the functions below as T and R for the
brothers.
____: 𝑑 = 450 − 55ℎ
____: 𝑑 = 45(ℎ − 1)
b. They want to eat dinner together, so they need
to know when both brothers will be the same
distance from Dallas. At what hour and how far
from Dallas should they expect to meet?
16) The function 𝑓(𝑥) = 𝑥 2 has undergone some
transformations. It has been shifted left 3 units,
stretched by a factor of 2, and reflected after it was
shifted 5 units down. Write the equation of the
resulting function.
18) Find the missing side in each right triangle.
20) In parallelogram 𝐽𝐾𝐿𝑀,
what is 𝑚∠𝐾?
19)
Directions: The following data represents the number of men and women passengers aboard the Titanic and whether or not the survived the
horrible disaster. First, rewrite the probability notation as a question, then use the table to find the probabilities.
Men
Women
Total
Survived
Did NOT
survive
Tot
146
296
442
659
106
765
805
402
1207
Example:
P(w): “What is the chance a random
person selected is a woman?”
𝑃(𝑤) =
21) P(s)
22) P(s│w)
23) P(w or s)
402
= 33.3%
1207
24)
25)
26)
27)
P(w or m)
P(ns│w)
P(m∩ns)
Historically, when populations
are at risk, society will protect
and save the women. Does this
data support that
generalization? Prove your
position with data.