STUDY GUIDE
GENERAL MATHEMATICS | UNIT 1
Introduction to Functions
Table of Contents
Introduction .................................................................................................................................... 3
Test Your Prerequisite Skills ........................................................................................................ 4
Objectives ...................................................................................................................................... 4
Lesson 1: Functions, Function Notations, and Equations
-
Warm Up! ........................................................................................................................... 5
-
Learn about It! ................................................................................................................... 6
-
Let’s Practice! ..................................................................................................................... 9
-
Check Your Understanding! ............................................................................................ 12
Lesson 2: Functions vs. Relations
-
Warm Up! ......................................................................................................................... 13
-
Learn about It! ................................................................................................................. 13
-
Let’s Practice! ................................................................................................................... 16
-
Check Your Understanding! ............................................................................................ 19
Lesson 3: Evaluating and Graphing Functions
-
Warm Up! ......................................................................................................................... 21
-
Learn about It! ................................................................................................................. 22
-
Let’s Practice! ................................................................................................................... 23
-
Check Your Understanding! ............................................................................................ 28
Lesson 4: Domain and Range of Functions
-
Warm Up! ......................................................................................................................... 30
-
Learn about It! ................................................................................................................. 30
-
Let’s Practice! ................................................................................................................... 35
-
Check Your Understanding! ............................................................................................ 38
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Challenge Yourself! ..................................................................................................................... 40
Performance Task ....................................................................................................................... 41
Wrap-up ....................................................................................................................................... 42
Key to Let’s Practice! .................................................................................................................... 43
References ................................................................................................................................... 45
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STUDY GUIDE
GRADE 11| GENERAL MATHEMATICS
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Table of Contents
UNIT 1
Introduction to Functions
You might come across one day when your mother
complains about the VAT or the Value Added Tax in your
electric and water bills. It might make her worry more the
fact that the VAT varies each month. You might find it
challenging to explain that the tax charged to consumers,
like your household, is a function of the amount of
consumption in a particular month.
Every day, you find situations where one quantity depends on
another. For instance, the scores you get in your exams depend
on the amount of work you put in studying. If your parents are
keen on your school progress, your allowance might depend on
your academic performance. A wage-earner’s income depends on
the number of hours he worked.
In business, profit depends on the total sales of a product.
Similarly, the interest of your savings depends on the amount you
deposited in a bank.
These are just some examples of the application of functions. In this unit, you will explore
various aspects of a function and its application to the real world.
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Test Your Prerequisite Skills
•
•
Plotting points in a Cartesian Plane
Evaluating algebraic expressions
Before you get started, answer the following items on a separate sheet of paper. This will
help you assess your prior knowledge and practice some skills that you will need in
studying the lessons in this unit. Show your complete solution.
A. Plot the following set of ordered pairs in the Cartesian plane and connect all the dots.
1. (2,0)
5. (2,8)
9. (6,4)
2. (2,2)
6. (4,8)
10. (7,8)
3. (2,4)
7. (6,8)
4. (2,6)
8. (4,4)
B. Evaluate each algebraic expression at the specified value of 𝑥.
1. 2𝑥 + 10 at 𝑥 = 18
2.
1
3𝑥+2
at 𝑥 = −1
4. (√𝑥)(4𝑥) at 𝑥 = 9
5. 2𝑥 2 + √𝑥 at 𝑥 = 8
3
3. 𝑥 2 − 7 at 𝑥 = 8
Objectives
At the end of this unit, you should be able to
•
define functions and distinguish them from equations;
•
distinguish functions from relations;
•
evaluate functions;
•
represent real-life situations using functions, including piecewise functions; and
•
identify the domain and range of different types of functions.
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STUDY GUIDE
Lesson 1: Functions, Function Notations, and Equations
Warm Up!
Rule Me Out!
Materials Needed:
pen and paper
Instructions:
1. This activity will be done by pair.
2. If the rule is given, complete the boxes with the correct numbers. Supply the
rule if it is missing.
I
N
P
U
T
RULE
Add 6 to the input
→
6
→
8
15 →
→
4
13 →
12 →
I
N
P
U
T
O
U
T
P
U
T
RULE
Subtract 10 from the input
→
2
→
34
→
3
→
16
→
5
→
8
O
U
T
P
U
T
RULE
I
N
P
U
T
12
−4
−16
8
4
−24
→
→
→
→
→
→
9
5
2
8
7
0
O
U
T
P
U
T
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STUDY GUIDE
Learn about It!
In an equation in two variables, 𝑥 and 𝑦, the variable 𝑦 may be expressed as 𝑓(𝑥) if every
value of 𝑥 corresponds to a single value of 𝑦. This is known as a function, where 𝑦 is a
function of 𝑥. It is usually denoted by 𝑦 = 𝑓(𝑥). Here, 𝑥 is the independent variable and 𝑦 is
the dependent variable.
The function 𝑓(𝑥) can be thought of as a “machine” that accepts values of 𝑥 as input and
produces values of 𝑦 as output. These outputs are also known as values of 𝑓(𝑥). In Warm
Up!, the rule acts as the function while the values are the input and the output.
Say, if you feed a tree trunk into a woodcutting machine, it will produce pieces of chopped
wood.
The tree trunks are the input, the machine is the function, and the chopped logs are the
output. Let’s take a look at an example as follows:
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In the equation of the line 𝑦 = 2𝑥 + 1, if 𝑥 = 1, 𝑦 = 3. If 𝑥 = 2, 𝑦 = 5 and if 𝑥 = 3, 𝑦 = 7. In
this equation, 𝑥 is the input, 2𝑥 + 1 is the function, and 𝑦 is the output.
Here are other ways of writing functions.
•
𝑓 (𝑥) = 3𝑥 + 1
𝑦 is written as a function of 𝑥, or 𝑓 (𝑥).
•
𝑥 → 3𝑥 + 1
The arrow is read as “is mapped to.”
•
𝑓 ∶ 𝑥 → 3𝑥 + 1
This method also names the function 𝑓.
•
𝑓 = {(𝑥, 𝑦)|𝑦 = 3𝑥 + 1}
Here, the function 𝑓 is written as a set.
Take note that 𝑓 and 𝑥 may be replaced with any other letters to represent problems
better. Thus, for example, the function that defines the area of a circle in terms of its
radius may be represented by 𝐴(𝑟). Similarly, 𝑣(𝑡) can be used to denote velocity as a
function of time.
Aside from the above methods of notation, some equations known as candidate
functions may also be used to represent functions. The equation 𝑦 = 3𝑥 + 1 , for
example, is a candidate function because we can substitute a value of 𝑥 to obtain a
corresponding value of 𝑦. But how do we distinguish between functions and equations?
Equations denote equality between two expressions. On the other hand, functions
denote relationships between two variables. A function always involves a relation
between a set of inputs and a set of outputs.
Piecewise Functions
Earlier, we showed an equation of a line, 𝑦 = 2𝑥 + 1. If we plot this equation in a
coordinate plane, we will generate the following graph, in which the line extends infinitely
for the values along the 𝑥-axis.
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Notice that the 𝑥-axis can actually be divided into intervals. Also, we can sketch parts of
different functions along each interval. For example, the following figure shows the graph
1
1
of 𝑓(𝑥) = 2 in the interval −2 < 𝑥 < 0, and the graph of 𝑓(𝑥) = − 2 elsewhere.
This is called a piecewise function. It can be represented by writing the appropriate
functions for each interval.
For the given figure, the piecewise function is defined as follows:
1
if 𝑥 < −2
2
1
𝑓(𝑥) =
if − 2 < 𝑥 < 0
2
1
𝑥>0
{− 2 if
−
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STUDY GUIDE
Let’s Practice!
Example 1: In a clothes dryer, what do you think is the input, the
function, and the output?
Solution:
Putting wet clothes in the dryer will cause the clothes to
dry. Hence, the wet clothes are the input, the drying
process is the function, and the results (the output) are
the dry clothes.
Try It Yourself!
Which of the following machines can be considered as a function?
a. Television set
b. Paper shredder
c. Electric fan
Example 2: Analyze the given chart below. Determine the input, the function, and the
output.
Solution:
𝒙
1
2
3
4
5
𝒚
4
5
6
7
8
The input are the 𝑥-values (1, 2, 3, 4, 5). The output are the 𝑦-values (4, 5, 6, 7,
8). Notice that if the value of 𝑥 is 1, the value of 𝑦 is 4. If 𝑥 is 2, 𝑦 is 5, and so
on. By observing the pattern, note that each input increases by 3 after “going
through” the function. Hence, the function is 𝑦 = 𝑥 + 3.
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STUDY GUIDE
Try It Yourself!
Using the graph shown, complete the table by
writing the inputs and outputs of the function.
Use integral values.
Input (𝒙)
Output (𝒚)
𝑦 =𝑥+2
Example 3: Consider the following equations:
a. 3𝑥 = 4𝑥 − 1
b. 𝑦 = 3𝑥 + 1
c. 2 + 5 = 6 + 1
Which of the given equations is a function?
Solution:
The only function among the three equations is 𝑦 = 3𝑥 + 1, because it
denotes a relationship between two variables 𝑥 and 𝑦.
Try It Yourself!
Tell whether each equation is a function or not.
a. 𝑓(𝑥) = 0
b. 𝑥 = 3𝑦 + 2
c. 3𝑚 = 𝑚 + 1
d. 𝑓(𝑥) = 𝑥 2 + 2
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STUDY GUIDE
Real-World Problems
Example 4: A game at a theme park offers prize money based on
the number of balls you can successfully shoot in
one minute.
•
0-7 successful shots: no prize
•
8-15 successful shots: ₱150
•
more than 15 successful shots: ₱200
Write a piecewise function 𝑀(𝑥) that relates the prize money offered to the
number of successful shots you make.
Solution:
The piecewise function that can represent the problem is
0
𝑀(𝑥) = { 150
200
if 0 ≤ 𝑥 ≤ 7
if 8 ≤ 𝑥 ≤ 15
if
𝑥 > 15
Try It Yourself!
In a T-shirt store, the manager gives a 5% discount for a
purchase of 1-10 pieces, 10% discount for 11-20 pieces,
and 15% discount for a purchase of 21 pieces and more.
Write a piecewise function that relates the discount to the
number of T-shirts purchased.
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STUDY GUIDE
Check Your Understanding!
1. Put a check mark (✓) on the blank if the item is considered a “function machine”. If
not, put a cross mark ().
___a.
Meat grinder
___f.
Sewing machine
___b.
Chain saw
___g.
Pencil sharpener
___c.
Computer
___h.
Refrigerator
___d.
Bread toaster
___i.
Paper cutter
___e.
Vacuum cleaner
___j.
Meat slicer
2. Tell whether each equation is a function or not.
a. 4 − 𝑦 = 𝑥
b. 𝐴(𝑟) = 𝜋𝑟 2
c. 4 = 12 − 8
d. 𝑓(𝑥) = 12𝑦
e. 𝑦 = 2
f.
g.
h.
i.
j.
5𝑥 + 4𝑦 = 9
34 − 2𝑥 = 14 + 2𝑦
𝑃(𝑐) = 𝑐 + 0.5𝑐
6𝑡 = 𝐺(𝑡) − 5
6−4+3= 𝑥
3. Write T beside the statement if the statement is true. Otherwise, write F for false
statements.
a. In the function 𝑦 = −6𝑥 + 4, 𝑥 is the dependent variable and 𝑦 is the
independent variable.
b. Functions can be modeled by the “input-process-output” machine.
c. 𝑓 = {(𝑥, 𝑦)|𝑦 = 𝑥} can be considered a function.
d. All equations are functions.
e. All functions are equations.
4. Write a piecewise function for the problem.
a. Mang Ernesto works as a candle maker. He has to produce more than 200
pieces of candles to earn ₱2 per candle. If the number of candles produced is
from 100 to 200, he will earn ₱1.50 per candle. He will then get ₱1 per candle for
the number of candles produced less than 100.
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STUDY GUIDE
Lesson 2: Functions vs. Relations
Warm Up!
Name That Young!
Materials Needed:
pen and paper
Instructions:
1. This activity should be done individually.
2. Refer to the columns below. On the left are animals’ names and on the right are the
names given to the animals’ young.
3. Draw an arrow to connect the name of an animal to its young.
Animal
Elephant
Cat
Penguin
Giraffe
Camel
Chicken
Young
Kitten
Calf
Chick
Learn about It!
The activity in Warm Up! illustrates relations and functions. As you make progress in this
lesson, you will be able to identify the relation shown in the previous activity.
Definition 2.1: A relation is a set of objects, such as numbers,
grouped with each other that may or may not
represent a pattern. It is simply a set of ordered pairs
that are arranged in an orderly manner.
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STUDY GUIDE
There are four types of relations of ordered pairs. Study the following.
One-to-one correspondence
Each value of 𝑥 is unique and is associated with a unique value of 𝑦.
Examples:
Observe that each number in the 𝑥-column is paired with a particular number in the 𝑦column.
Many-to-one correspondence
Two or more values of 𝑥 are associated with the same value of 𝑦.
Example:
In this case, the numbers 1, 2, 3 (𝑥-values) are paired with the same 𝑦-value (4).
One-to-many correspondence
Some values of 𝑥 are associated with more than one value of 𝑦.
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STUDY GUIDE
Example:
Observe that the number 2 (𝑥-value) is paired with different 𝑦-values (2, 4, 6, 8).
Many-to-many correspondence
Some values of both 𝑥 and 𝑦 are associated with more than one value of their
counterpart.
Example:
One-to-one and many-to-one relations are both considered functions.
Definition 2.2: A function is a special kind of relation in which
no two distinct ordered pairs have the same first
element.
Since relations are composed of ordered pairs, we may plot the corresponding points on
the Cartesian plane. Similarly, a relation or function defined by an equation can be
represented by the graph of the said equation.
The vertical line test can be used to determine if a graph represents a function. If a
vertical line is drawn on any part of the graph of a function, the line will intersect one
point on the graph. On the other hand, if we can draw a vertical line that intersects a given
graph at more than one point, then that graph represents a relation that is not a function.
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STUDY GUIDE
This is because there are points with the same 𝑥-coordinate (one value of the
independent variable) but different 𝑦-coordinates (multiple values of the dependent
variable). This is an example of a one-to-many relation.
Let’s Practice!
Example 1: What type of relation best describes the following?
a. Students and their ID numbers
b. A family with 5 members living in the same house
c. A class adviser to her students
d. Students and classes
Solution:
a.
In a school, a student owns exactly one ID. Thus, this is an example of one-toone correspondence.
b.
This is a many-to-one relation where family members correspond to one
house.
c.
One class adviser corresponds to many students in a class. So, this is a oneto-many type of relation.
d.
This is a many-to-many relation. Each student attends multiple classes. At the
same time, each class has many students.
Try It Yourself!
Identify the type of relation that best describes each situation.
a. Students in a class and their birthdays
b. Countries and their national flags
c. Books and authors
d. Mothers and daughters/sons
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STUDY GUIDE
Example 2: Write all the ordered pairs in the figure and determine
the type of relation depicted by the given figure.
Solution:
Let us list all the ordered pairs:
(1, 5), (1, 6), (2, 8), (3, 8)
This is a many-to-many type of relation. Note that an element in set A, like 1,
is associated with multiple elements in set B (5 and 6). Similarly, an element
in set B, like 8, is associated with multiple elements in set A (2 and 3).
Try It Yourself!
List all the ordered pairs in the given figure and determine
the type of relation.
Example 3: Tell whether each graph represents a function or not.
a
b
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STUDY GUIDE
Solution:
a.
The graph represents a function because when a vertical line is drawn
anywhere on the graph, it will pass through the graph at exactly one point.
b.
The graph does not represent a function. Note that a vertical line is already
drawn. Observe that the line crosses the graph at more than one point. Thus,
the graph failed the vertical line test.
Try It Yourself!
Graph the values on a coordinate plane and determine if it
represents a function or a relation.
𝒙
−2
−1
0
1
2
𝒚
6
0
−2
0
6
Real-World Problems
Example 4: The weight of Mrs. Lazaro’s baby is being monitored
by their family pediatrician. She constructed a table of
the baby’s weight in kilograms with respect to the
baby’s age in weeks. Will the ordered pairs from the
pediatrician’s data represent a function or a relation?
Explain your answer.
Solution:
The ordered pairs in the pediatrician’s data will form a function because
there will be a one-to-one or many-to-one correspondence in the baby’s
weight and the baby’s age. There will be a one-to-one correspondence if the
baby’s weight is unique each week the weight is recorded. On the other
hand, there will be a many-to-one correspondence if there are weeks when
the baby’s weights are the same. In any case, however, the data will
represent a function.
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STUDY GUIDE
Example 5: Mrs. Salazar collects data of her son’s progress in
Math for four quarters of a school year. She then
constructs a table with two variables; the four
quarters of the school year and her son’s Math grade
for the specific quarter. Do the data represent a
function or a relation? Explain your answer.
Solution:
The data represent a function because there is a one-to-one correspondence
(i.e., the grades each quarter is unique) or a many-to-one correspondence
(i.e., there are quarters where the grades are the same). In any case, the data
represent a function.
Try It Yourself!
The HR manager of a publishing house found out that Ms.
Jaime is listed as author of different books. Does this show a
function or a relation? Explain your answer.
Check Your Understanding!
1. Draw a diagram of each set of ordered pairs and determine if each set represents a
one-to-one, one-to-many, many-to-one, or many-to-many correspondence.
a.
b.
c.
d.
e.
(2, 3), (3, 2), (2, 5), (5, 2)
(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)
(1, 4), (2, 4), (3, 4), (4, 4)
(1, 𝑎), (1, 𝑏), (1, 𝑐), (1, 𝑑)
(3, −2), (3, 2), (4, −2), (5, 2)
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2. Graph the following points on a coordinate plane and determine if each graph is a
function or a relation.
a.
𝒙
−2
𝒚
−1
3
0
2
1
2
2
6
𝒙
−2
−1
𝒚
−8
−5
0
2
1
1
2
4
6
b.
3. Analyze and answer the problem below.
The head of the accounting department collects data of the employees’ salary and
the number of overtime hours they rendered. If the data are used as variables and
plotted in a chart, would the chart show a function or a relation? Explain your
answer.
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STUDY GUIDE
Lesson 3: Evaluating and Graphing Functions
Warm Up!
Let’s Race to More!
Materials Needed:
2 colored pen and paper
Instructions:
1. This activity will be done by pair.
2. The first player will introduce a subject and the second player will name as many
examples of the subject as he or she can.
Example: Country starting with the letter C (Canada, Costa Rica, China, ...)
3. The number of examples the first player was
able to name will then be plotted on a
graphing paper given on the right. This will be
Round 1 for player 1.
4. The second player will introduce another
subject while the first player will enumerate
examples. This will be Round 1 for player 2.
5. The result will also be plotted on the same
graph. Be sure to use a different color of pen
to distinguish one from the other.
6. The game can be played in minimum of five
rounds.
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STUDY GUIDE
Learn about It!
One of the most important processes involving functions is to evaluate them for given
values of the variables. Just like algebraic expressions, functions are evaluated by
substituting the given values of the variables and simplifying the resulting expression. If
we evaluate a function 𝑓(𝑥) at a certain value, say 𝑥 = 0, the resulting value of 𝑓(𝑥) is
denoted by 𝑓(0). Evaluating functions at given points is also a necessary skill in graphing
functions.
Graphing Functions
In Warm Up!, we have reviewed how to plot points in a Cartesian plane. The skill in
accurately plotting points in a coordinate plane is important in graphing functions.
You can graph any equation or function using a table of values. A table of values is a
graphic organizer or chart that helps you determine two or more points that can be used
to create your graph.
Suppose we want to sketch the graph of 𝑦 = 2𝑥 + 1, we can follow the given steps.
Solution:
Step 1:
Tabulate some arbitrary values of 𝑥.
𝒙
−2
𝒚
−1
0
1
2
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STUDY GUIDE
Step 2:
To obtain the corresponding values of 𝑦, evaluate the function at the
chosen values of 𝑥.
𝒙
−2
−1
0
1
2
Step 3:
𝒚
−3
−1
1
3
5
Plot the ordered pairs on the Cartesian plane.
Let’s Practice!
Example 1: Evaluate the function 𝑓 (𝑥) = 3𝑥 + 1 at (a) 𝑥 = 2 and (b) 𝑥 = 3.
Solution:
a.
Substitute 𝑥 = 2 and simplify.
𝑓(𝑥) = 3𝑥 + 1
𝑓(2) = 3(2) + 1
𝑓(2) = 6 + 1
𝑓(2) = 7
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STUDY GUIDE
b.
Substitute 𝑥 = 3 and simplify.
𝑓(𝑥) = 3𝑥 + 1
𝑓(3) = 3(3) + 1
𝑓(3) = 9 + 1
𝑓(3) = 10
Try It Yourself!
Evaluate 𝑓(𝑥) = −4𝑥 + 2 at (a) 𝑥 = 0 and (b) 𝑥 = 2.
𝑥+2
if 𝑥 < 2
Example 2: Evaluate 𝑓(𝑥) = { 2𝑥 2 − 5 if 𝑥 = 2 when (a) 𝑥 = 0, (b) 𝑥 = 2, and (c) 𝑥 = 4.
√𝑥 + 1 if 𝑥 > 2
Solution:
a.
Since 0 < 2, we use the first expression. Substitute 0 in the expression, then
simplify.
𝑓(𝑥) = 𝑥 + 2
𝑓(0) = 0 + 2
𝑓(0) = 2
b.
Since 2 = 2, we use the second expression. Substitute 2 in the expression,
then simplify.
𝑓(𝑥) = 2𝑥 2 − 5
𝑓(2) = 2(22 ) − 5
𝑓(2) = 2(4) − 5
𝑓(2) = 8 − 5
𝑓(2) = 3
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STUDY GUIDE
c.
Since 4 > 2, we use the third expression. Substitute 4 in the expression, then
simplify.
𝑓(𝑥) = √𝑥 + 1
𝑓(4) = √4 + 1
𝑓(4) = √5
Try It Yourself!
𝑥−3
if 𝑥 < 1
2
𝑥
+
4
if
𝑥 = 1 when (a) 𝑥 = 1, (b) 𝑥 = 3, and (c) 𝑥 = −1.
Evaluate 𝑓(𝑥) = {
√𝑥 + 2 if 𝑥 > 1
Example 3: Sketch the graph of the function 𝑓(𝑥) = 𝑥 2 + 2𝑥 + 1.
Solution:
Step 1:
Step 2:
Tabulate some arbitrary values of 𝑥.
𝒙
−2
𝒚
−1
0
0
1
1
4
2
9
1
To obtain the corresponding values of 𝑓(𝑥), evaluate the function at the
chosen values of 𝑥.
𝒙
−2
𝒇(𝒙)
−1
0
1
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STUDY GUIDE
Step 3:
0
1
1
4
2
9
Plot the ordered pairs in the Cartesian plane.
Try It Yourself!
Sketch the graph of the function 𝑓(𝑥) = 3𝑥 2 + 𝑥 − 1.
Real-World Problems
Example 4: Roberto receives a monthly salary of ₱12,000. On
top of his monthly salary, he also receives a
commission of ₱150 for every cellphone that costs
₱15,000 and above that he sells. How much will
Roberto receive in a month if he was able to sell 20
cell phones that cost ₱15,000 and above?
Solution:
Let 𝑐 be the number of cellphones sold and 𝑅(𝑐) be Roberto’s monthly salary.
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STUDY GUIDE
Substitute the given values in the function.
𝑅(𝑐) = 150𝑐 + 12 000
𝑅(20) = 150(20) + 12 000
𝑅(20) = 3 000 + 12 000
𝑅(20) = 15 000
Thus, Roberto will receive ₱15,000 in a month if he was able to sell 20
cellphones that cost ₱15,000 and above.
Example 5: The cost of producing 𝑥 gadget by a company
is given (in pesos) by the function
𝑓(𝑥) = 1 500𝑥 + 7 000. What is the cost of
producing 50 gadgets? 100 gadgets?
Solution:
We can get the cost of producing 50 gadgets
by substituting 50 in the function.
𝑓(𝑥) = 1 500𝑥 + 7 000
𝑓(50) = 1 500(50) + 7 000
𝑓(𝑥) = 82 000
To find the cost of producing 100 gadgets, substitute 100 in the given
function.
𝑓(𝑥) = 1 500𝑥 + 7 000
𝑓(100) = 1 500(100) + 7 000
𝑓(100) = 157 000
Therefore, the cost of producing 50 gadgets is ₱82,000 while the cost of
producing 100 gadgets is ₱157,000.
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STUDY GUIDE
Try It Yourself!
Mr. Gozon starts his journey with 50 liters in the tank of
his car. The car burns 1 liter for every 12 kilometers. If the
amount of gasoline decreases linearly, write a linear
function to represent the number of liters left in the tank
after a journey of 𝑥 kilometers. How much gas is left after
traveling 132 km?
Check Your Understanding!
1. Evaluate the given functions at the specified values.
1
a. 𝑓(𝑥) = 2 𝑥 − 1
i.
ii.
iii.
iv.
v.
𝑓(0)
𝑓(2)
𝑓(−1)
𝑓(4)
𝑓(−2)
b. 𝑔(𝑥) = 3𝑥 + 5
i.
𝑔(−3)
ii.
𝑔(−1)
iii.
𝑔(1)
iv.
𝑔(3)
v.
𝑔(5)
2. Given the graph of the function, find the specified
values.
a.
i.
ii.
iii.
𝑓(−1)
𝑓(1)
𝑓(0)
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STUDY GUIDE
b.
i.
ii.
iii.
𝑓(1)
𝑓(−1)
𝑓(0)
3. Sketch the graph of each function.
a. 𝑓(𝑥) = −𝑥 2 + 2.
b. 𝑓(𝑥) = 2𝑥 2 − 3𝑥 + 1
c. 𝑓(𝑥) = 𝑥 3 − 3
4. Solve the problem.
A 100-liter tank full of oil is being drained at a constant rate of 5 liters per minute.
a. Assuming that the draining starts at time 𝑡 = 0, write a linear function 𝑉 for the
number of liters in the tank after 𝑡 minutes.
b. How many liters are in the tank after 15 minutes?
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STUDY GUIDE
Lesson 4: Domain and Range of Functions
Warm Up!
Region VI, Here We Come!
Materials Needed:
pen and paper
Instructions:
1. This activity should be done individually.
2. List all the provinces of Region VI – Western Visayas Region including the capital
cities.
3. Use the format below in listing:
(Province, Capital City)
Learn about It!
The activity in Warm Up! illustrates two important properties of a function—the domain
and range. Note that we have restricted our activity to provinces in the Western Visayas
Region. In other words, the only input we can use in our function are Region VI provinces.
For instance, we do not expect to use La Union in the activity because it is not part of
Western Visayas. In this activity, we have set Region VI provinces as the domain of the
function. Observe also that when we set our domain to be these provinces, our capital
cities also became restricted. The only choices we have for capital cities are, of course,
cities that are within the provinces we have specified. These cities are the range of our
function. They are the only possible outputs.
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STUDY GUIDE
It is often useful to determine the set of all possible values of 𝑥 and 𝑦 in a function.
These are known as the domain and the range of the function.
Definition 4.1: The domain of a function is the set of
all values of the independent variable
𝑥 that have corresponding values of
the dependent variable 𝑦. It contains
all values that go into the function.
Definition 4.2: The range of a function is the set of all
values of 𝑦 that can be obtained from
the possible values of 𝑥. It contains all
possible values of the function.
Linear Function
If a function is defined in the form of a linear equation, it is called a linear function. It has a
degree of 1 and its graph is a straight line. The domain of any linear function is the set of
real numbers. No matter what real number we substitute to 𝑥, the corresponding value of
𝑓(𝑥) is also a real number.
Similarly, the range of any linear function is also the set of real numbers. No matter what
value we choose for 𝑓(𝑥), there is always a value of 𝑥 that can be used to obtain the
chosen value of 𝑓(𝑥).
Quadratic Function
If a function is defined in the form of a quadratic equation, it is known as a quadratic
function. It has a degree of 2 and its graph is a parabola.
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STUDY GUIDE
The domain of any quadratic function is the set of real numbers. In this case, all real
numbers can be squared, so all real numbers can be used as values of 𝑥.
To determine the range of the function, remember that the square of a real number is
always a nonnegative real number. For instance, (−22 ) = 4, 02 = 0, and 22 = 4. Thus, the
range is the set of all nonnegative real numbers. Generally, the range of a quadratic
function always contains restrictions like this. If 𝑦 = 𝑓(𝑥) is a quadratic function, then its
range can be written in either of the forms 𝑦 ≥ 𝑐 or 𝑦 ≤ 𝑐 , where 𝑐 is a real number.
Thus, the domain of 𝑓(𝑥) is 𝑥 ∈ ℝ and the range of 𝑓(𝑥) is 𝑦 ≥ 0.
Polynomial Function
If a function is defined in the form of a polynomial equation whose degree is greater than
2, it is called a polynomial function.
If the degree of a polynomial function is odd, then its domain and range are both equal to
the set of real numbers.
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STUDY GUIDE
Using a graph of a polynomial function above, if we draw a vertical line anywhere on this
Cartesian plane, it will always intersect the graph of the function. This means that all real
numbers are used as 𝑥-coordinates of points on the graph. Hence, all real numbers are
part of the domain. Similarly, any horizontal line drawn on the Cartesian plane will always
intersect the graph of the function as well. This implies that all real numbers are also used
as 𝑦-coordinates. Thus, all real numbers are part of the range.
Therefore, the domain of 𝑓(𝑥) is 𝑥 ∈ ℝ and the range of 𝑓(𝑥) is 𝑦 ∈ ℝ. This is the case for
every polynomial function whose degree is 3, 5, 7 or any other odd number.
Rational Function
If a function is defined in the form of a rational equation, it is called a rational function. It
is the ratio of two polynomials.
Let us take for example the function 𝑓(𝑥) =
1
. Unlike the functions in the previous
𝑥−2
example, the domain of a rational function has restrictions if the variable 𝑥 can be found
in the denominator because this can possibly result in a denominator of 0. This restriction
can be found by setting the denominator equal to 0 and solving for 𝑥. In this case, we
have
𝑥−2=0
𝑥=2
Thus, the domain of 𝑓(𝑥) is 𝑥 ≠ 2, because a denominator of zero will make the function
undefined.
To find the range, notice that the numerator of 𝑓(𝑥) is nonzero and does not contain the
variable 𝑥. Hence, no matter what value of 𝑥 we choose, we will never obtain 𝑓(𝑥) = 0. This
is because there is no divisor that can divide 1 to produce a result of 0. This implies that
the range of 𝑓(𝑥) is 𝑦 ≠ 0.
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STUDY GUIDE
The graph of 𝑓(𝑥) =
1
𝑥−2
is shown above. Observe that it does not intersect the lines 𝑥 = 2
and 𝑦 = 0, which indicates that these are indeed restrictions in the domain and range of
the function.
Radical Function (Square Root Function)
If a function is defined in the form of an equation that contains radical expressions, it is
called a radical function.
Let’s find the domain and range of 𝑓(𝑥) = √𝑥 − 3. The general restriction for the domain of
a square root function stems from the fact that the square root of a negative number is
not a real number. Thus, to find the domain of 𝑓(𝑥), we assume that the radicand is
nonnegative (greater than or equal to zero) and solve for 𝑥.
𝑥 −3 ≥0
𝑥 ≥3
Thus, the domain of 𝑓(𝑥) is ≥ 3 .
To find the range of 𝑓(𝑥), recall that the principal square root of a radical expression is
always nonnegative. Generally, the range of a radical function is 𝑦 ≥ 0 because of the
definition of the principal square root.
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STUDY GUIDE
In the above graph of 𝑓(𝑥) = √𝑥 − 3, we can observe that the curve does not contain any
points to the left of 𝑥 = 3 or below 𝑦 = 0, which indicates that our restrictions are correct.
Let’s Practice!
Example 1: Find the domain and range of the function 𝐹 = {(−3, 2), (0, 1), (4, 2), (5, 2)}.
Solution:
If a function is expressed as a set of ordered pairs, then its domain is the set
of all 𝑥-coordinates while its range is the set of all 𝑦-coordinates. In this case,
the domain of 𝐹 is {−3, 0, 4, 5} and the range of 𝐹 is {1, 2}.
Try It Yourself!
Find the domain and range of the function 𝐺 = {(1, 0), (2, −3), (6, 5), (0, −1)}.
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STUDY GUIDE
Example 2: Find the domain and range of the function 𝑓(𝑥) = 𝑥 + 1.
Solution:
Since the function is a linear function, the domain is the set of real numbers.
Likewise, the range is also the set of real numbers. Therefore, the domain of
the function 𝑓(𝑥) = 𝑥 + 1 is 𝑥 ∈ ℝ and the range is 𝑦 ∈ ℝ.
Try It Yourself!
What is the domain and range of the 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 2?
Example 3: Find the domain and range of 𝑦 = |𝑥|.
Solution:
If we try to graph the function, it will look like the following graph.
Based on the graph, the line extends far up to the left and far up to the right.
Thus, the domain of the function is all real numbers and the range of the
function is 𝑦 ≥ 0.
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STUDY GUIDE
Try It Yourself!
Find the domain and range of the function
represented by the graph.
Real-World Problems
Example 4: The height of a ball that is dropped from a 100-ft
building at any time 𝑡 is given by
ℎ(𝑡) = −16𝑡 2 + 100, where 𝑡 is time in seconds.
What are the domain and the range of this
function?
Solution:
To get the domain of the function, we shall set the
function to zero. Thus,
0 = −16𝑡 2 + 100
16𝑡 2 = 100
100
𝑡2 =
16
𝑡 = ±2.5
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STUDY GUIDE
Hence, the domain of the function is −2.5 ≤ 𝑡 ≤ 2.5. However, the 𝑡
represents time and time should be a nonnegative number. Thus, the
domain of the given function is 0 < 𝑡 ≤ 2.5.
The range of the function is also a nonnegative number which cannot exceed
100. Therefore, the range of the function is 0 < ℎ ≤ 100.
Try It Yourself!
Patrick walks at an average rate of 5 km/h. The distance that he walks
can be represented by 𝑑(𝑡) = 5𝑡, where 𝑡 is the time in hours. What are
the domain and range of the given function? Explain your answer.
Check Your Understanding!
1. Identify the type of function in each item.
a.
b.
c.
d.
𝑓(𝑥) = 3𝑥 2 + 1
𝑓(𝑥) = −2𝑥 3 + 2𝑥
𝑓(𝑥) = 5𝑥 − 3
𝑓(𝑥) = √𝑥 + 6
e. 𝑓(𝑥) =
1
𝑥+2
2. Find the domain and range of the following functions:
a. 𝑓(𝑥) = 4𝑥 − 2
b. 𝑓(𝑥) = 𝑥 2 + 3
c. 𝑓(𝑥) = 𝑥 3 + 2𝑥 2 − 3𝑥 + 1
2
d. 𝑓(𝑥) = 𝑥−1
e. 𝑓(𝑥) = √𝑥 + 4
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STUDY GUIDE
3. Determine the domain and range from the given graphs.
a.
b.
c.
d.
4. Find the domain and range of the function represented by the problem below.
A missile’s trajectory was calculated to have followed ℎ(𝑡) = −4𝑡 2 + 30𝑡 + 12, where
ℎ is the missile’s altitude (in kilometres) and 𝑡 is the time elapsed (in minutes).
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STUDY GUIDE
Challenge Yourself!
1. Examine the more complicated buttons on a scientific calculator like log 𝑥 , 𝑥 2 , exp,
sqrt, and sin. Construct a table of values for each of these functions.
2. Consider the following set of ordered pairs:
{(4,60), (2,30), (5,75), (10,150), (8,120)}
Arrange them in increasing order using the first element, then predict the most
likely ordered pairs whose first elements are 1, 3, 6, 7, and 9 based on the pattern
formed.
3. How is the graph of 𝑓(𝑥 + 3) − 2 compared to the graph of 𝑓(𝑥)? You may use
illustration to support your answer.
4. Choose one function. Represent this function using an equation, a table of values,
and a graph. Determine the domain and the range of the function.
5. Graph the equation 𝑥 2 + 𝑦 2 = 1. Does the graph represent a function? Explain your
answer.
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STUDY GUIDE
Performance Task
You are an aspiring entrepreneur who wants to put up a T-shirt business. You are tasked
to make a proposal that contains the following, among others:
1. A formula that will represent your profit as a function of cost
Determine the number of T-shirts where you will have a breakeven, a loss, and a
profit.
2. A table of values and a graph
3. The initial capital you will be needing to start off the business
You are to present your output to a potential investor.
Performance Task Rubric
Criteria
Accuracy
Delivery
Below
Expectation
(0–49%)
The computations
are erroneous and
do not show the
use of related
mathematical
concepts.
There is no
explanation of the
solution. The
explanation
cannot be
understood, or is
unrelated to the
task. There is no
Needs
Improvement
(50–74%)
The
computations are
erroneous and
show some use
of related
mathematical
concepts.
There is an
incomplete
explanation; it is
not clearly
represented.
There is some
use of
appropriate
Successful
Performance
(75–99%)
Exemplary
Performance
(99+%)
The computations
are accurate and
show the use of
related
mathematical
concepts.
The computations
are accurate and
show a wise use of
related
mathematical
concepts.
There is a clear
explanation and
appropriate use
of accurate
mathematical
representation.
There is effective
use of
There is a clear and
effective explanation
of the solution. All
steps are included
so the audience
does not have to
infer how the task
was completed.
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STUDY GUIDE
use or
inappropriate use
of mathematical
representation
and terminology
for the task.
mathematical
representation
and terminology
for the task.
mathematical
terminology.
Mathematical
Justification
It shows no
understanding of
the problem, The
solution
addresses none of
the mathematical
components
required to solve
the task.
It shows some
understanding of
the required
mathematical
knowledge. The
solution
addresses some
of the
mathematical
components
presented in the
task.
It shows
understanding of
required
mathematical
knowledge. The
solution
addresses most
of the
mathematical
components
presented in the
task.
Submission
of Proposal
The proposal is
submitted more
than five days late.
The proposal is
submitted four to
five days late.
The proposal is
submitted two to
three days late.
Mathematical
representation is
actively used as a
means of
communicating
ideas. Precise and
appropriate
mathematical
terminology is used.
It shows in-depth
understanding of the
required
mathematical
knowledge in
probability. The
solution completely
addresses all
mathematical
components
presented in the
task.
The proposal is
submitted on or
before the target
date.
Wrap-up
Looking For
Comparing functions vs.
equations
What to Do
Analyze input and output.
Comparing functions vs.
•
Check correspondence: One-to-one or many-to-one.
relations
•
Use the Vertical Line test.
Evaluating functions
Solving word problems
Substitute the value of 𝑥 and simplify.
Construct a working function and solve for the unknown.
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STUDY GUIDE
Key Terms/Formulas
Key Term
Description
Equation
denotes equality between two expressions
complete set of all possible values of the
Domain of a function
independent variable/input
a relation between a set of inputs that is related to
Function
exactly one output
Piece-wise function
a function defined by multiple subfunctions
Range of a function
set of values that are produced by that function
Relation
any set of ordered pairs (𝑥, 𝑦)
Key to Let’s Practice!
Lesson 1
1. Paper shredder
2.
3.
Input (𝒙)
Output (𝒚)
0
2
1
3
2
4
3
5
a. function
b. function
c. not a function
d. function
4.
5% if 1 ≤ 𝑥 ≤ 10
𝐷(𝑥) = {10% if 11 ≤ 𝑥 ≤ 20
15% if 𝑥 > 21
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STUDY GUIDE
Lesson 2
1. a. One-to-one or many-to-one
b. One-to-one
c. Many-to-many
d. One-to-one or one-to-many
2. Many-to-many
(1, 𝑎), (1, 𝑏), (2, 𝑏), (3, 𝑐), (4, 𝑐), (4, 𝑑)
3. Function
4. Relation
Lesson 3
1. 𝑓(0) = 2, 𝑓(2) = −6
2. a. 5
b. √5
c. −4
3.
𝑥
4. 𝑓(𝑥) = 50 − 12 ; 39 liters
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STUDY GUIDE
Lesson 4
1. Domain={1, 2, 6, 0}
Range={0, −3, 5, −1}
2. 𝑥 ∈ ℝ, 𝑦 ≥ 1
3. 𝑥 ∈ ℝ, 𝑦 ≥ −4
4. Domain: 𝑡 > 0, range: 𝑑(𝑡) > 0
References
Purplemath. “Functions versus Relations.” Accessed April 25, 2018.
http://www.purplemath.com/ modules/fcns.htm
Khan Academy. “Linear Equations and Functions.” Accessed April 25, 2018.
https://www.khanacademy. org/math/cc-eighth-grade-math/cc-8th-linear-equationsfunctions/
Simmons, George F. Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry.
Eugene, Oregon 97401: Resource Publications, 2003
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