General Mathematics
Quarter 1 – Module 2:
Evaluating Functions
General Mathematics
Alternative Delivery Mode
Quarter 1 – Module 2: Evaluating Functions
First Edition, 2020
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General Mathematics
Quarter 1 – Module 2:
Evaluating Functions
Introductory Message
For the facilitator:
Welcome to Grade 11 General Mathematics Alternative Delivery Mode (ADM) Module
on Evaluating Functions!
This module was collaboratively designed, developed and reviewed by educators from
public institutions to assist you, the teacher or facilitator in helping the learners
meet the standards set by the K to 12 Curriculum while overcoming their personal,
social, and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and independent
learning activities at their own pace and time. Furthermore, this also aims to help
learners acquire the needed 21st century skills while taking into consideration their
needs and circumstances.
In addition to the material in the main text, you will also see this box in the body of
the module:
Notes to the Teacher
This contains helpful tips or strategies that
will help you in guiding the learners.
As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to manage
their own learning. Furthermore, you are expected to encourage and assist the
learners as they do the tasks included in the module.
For the learner:
Welcome to the General Mathematics Grade 11 Alternative Delivery Mode (ADM)
Module on Evaluating Functions!
The hand is one of the most symbolized part of the human body. It is often used to
depict skill, action and purpose. Through our hands we may learn, create and
accomplish. Hence, the hand in this learning resource signifies that you as a learner
is capable and empowered to successfully achieve the relevant competencies and
skills at your own pace and time. Your academic success lies in your own hands!
This module was designed to provide you with fun and meaningful opportunities for
guided and independent learning at your own pace and time. You will be enabled to
process the contents of the learning resource while being an active learner.
This module has the following parts and corresponding icons:
iii
What I Need to Know
This will give you an idea of the skills or
competencies you are expected to learn in the
module.
What I Know
This part includes an activity that aims to
check what you already know about the
lesson to take. If you get all the answers
correct (100%), you may decide to skip this
module.
What’s In
This is a brief drill or review to help you link
the current lesson with the previous one.
What’s New
In this portion, the new lesson will be
introduced to you in various ways such as a
story, a song, a poem, a problem opener, an
activity or a situation.
What is It
This section provides a brief discussion of the
lesson. This aims to help you discover and
understand new concepts and skills.
What’s More
This comprises activities for independent
practice to solidify your understanding and
skills of the topic. You may check the
answers to the exercises using the Answer
Key at the end of the module.
What I Have Learned
This
includes
questions
or
blank
sentence/paragraph to be filled in to process
what you learned from the lesson.
What I Can Do
This section provides an activity which will
help you transfer your new knowledge or skill
into real life situations or concerns.
Assessment
This is a task which aims to evaluate your
level of mastery in achieving the learning
competency.
Additional Activities
In this portion, another activity will be given
to you to enrich your knowledge or skill of the
lesson learned. This also tends retention of
learned concepts.
Answer Key
This contains answers to all activities in the
module.
At the end of this module you will also find:
iv
References
This is a list of all sources used in developing
this module.
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.
We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it!
v
What I Need to Know
This module was designed and written with you in mind. It is here to help you master
the key concepts of functions specifically on evaluating functions. The scope of
this module permits it to be used in many different learning situations. The language
used recognizes the diverse vocabulary level of students. The lessons are arranged
to follow the standard sequence of the course. But the order in which you read them
can be changed to correspond with the textbook you are now using.
After going through this module, you are expected to:
1. recall the process of substitution;
2. identify the various types of functions; and
3. evaluate functions.
What I Know
Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.
1. Which of the following is a polynomial function?
a. f ( x) = 2 x 2 − 10 x + 7
3
c. p ( x) = x − 7
b. g ( x) = 4 x 2 − 3 x + 8
d.
s ( x) = 2 m − 1
2. What kind of function is being illustrated by f ( x) = 2 x 3 − 3 x + 5 ?
a. Rational Function
c. Greatest Integer Function
b. Constant Function
d. Absolute Value Function
3. Find the function value given h( x) = 17 + 8 x of x = 4d .
c. 17 + 32d
a. 17 − 32d
d. 17 + 32d
4. Which of the following shows a logarithmic function?
b. 17 − 32d 2
a.
f ( x) = 8 x 3 + 8
b.
f ( x) = log 9 81
2
x
c. f ( x) = 3 − 6
f ( x) = x − 1 − 8
d.
5. Find the function value given h( x) = 7 x − 11 , if x = 8m + 3 .
a. 56m + 10
2
c. 56m + 10
b. 56m − 10
2
d. 56m − 10
1
6. Which of the following is the value of the function f ( x) = 3 x 2 − 15 x + 5 + 3 given
x = 3?
a. 25
b. 16
c. 19
d. 10
7. Evaluate the function h( x) = x + 31 given x = 2.5.
a. 34
b. -34
c. -33
d. 33
8. Give the value of the of the function c( x) = 5 x 3 − 18 at c(3) .
a. 117
b. 27
c. 153
d. 63
9. Evaluate: h( x) = 5 x 2 − 8 x + 12 given x = 5.
a. 22
b. 145
c. 97
d. -3
10. Find the value of the function h( x) = 5 x − 4 if x = 6 .
2
a.
80
16
c.
b. 2 19
d. 4
11. Evaluate the function f ( x) = 3x − 5 x + 2 given x = 2 x + 5 .
2
a. 12 x 2 + 50 x + 52
2
c. 12 x − 50 x + 52
b. 12 x 2 + 65 x + 77
2
d. 12 x − 65 x + 77
12. Given h( x) =
2x 2 − 5
, determine h(5).
3
a. -15
b. −
c. 15
5
d. 3
5
3
13. Evaluate the function k ( x) = 5 x if x =
a.
3
b.
5
c.
25
d.
14. Given g ( x) =
a.
2
.
3
3
25
2 x 2 − 3x + 7
, determine g ( 2) .
3x − 4
9
2
b. −
5
8
c. 7
9
2
d.
−
8
7
15. For what values of x can we not evaluate the function f ( x) =
a. ±4
b. ±3
c. ±2
d. ±1
2
3x + 7
?
x2 − 4
Lesson
1
Evaluating Functions
Finding the value of “x” for most of the students is what Mathematics is all about.
Sometimes, it seems to be a joke for the students to evaluate an expression, like
what is shown by the illustration.
Find x.
Here it is!
X
6
8
If you want to learn how to find the value of “y”, well then, you are in the right
page. WELCOME to your second module!
What’s In
Before we begin, let’s go back to the time when you first encounter how to evaluate
expressions.
Do you still remember?
Given the following expressions, find its value if x = 3.
1. x − 9
2. 3x + 7
3. x 2 + 4 x − 10
4.
2 x 2 − 6 x + 26
5.
3x 2 − 6
3
We have learned that, in an algebraic expression, letters can stand for numbers.
And to find the value of the expression, there are two things that you have to do.
1. Replace each letter in the expression with the assigned value.
First, replace each letter in the expression with the value that has been
assigned to it. To make your calculations clear and avoid mistakes, always
enclose the numbers you're substituting inside parentheses. The value that's
given to a variable stays the same throughout the entire problem, even if the
letter occurs more than once in the expression.
However, since variables "vary", the value assigned to a particular variable can
change from problem to problem, just not within a single problem.
2. Perform the operations in the expression using the correct order of
operations.
Once you've substituted the value for the letter, do the operations to find the
value of the expression. Don't forget to use the correct order of operations: first
do any operations involving exponents, then do multiplication and division, and
finally do addition and subtraction!
If in the activity above, you do the same process in order to arrive with these answers,
then, this module seems to be very easy to you.
Solutions:
Given the following expressions, find its value if x = 3.
1. x − 9
Since x = 3, we just replaced
x by 3 in the expression,
then subtract by 9.
= x−9
= (3) − 9
= −6
2. 3x + 7
= 3x + 7
= 3(3) + 7
=9+7
= 16
Following the steps, we just
replace x by 3, multiply it by the
numerical coefficient 3, then add
7
4
3.
x 2 + 4 x − 10
After replacing x by 3, we
get the squared of 3 which
is 9, add it to the product
of 4 and 3, then lastly, we
subtracted 10 from its
sum.
= x 2 + 4 x − 10
= (3) 2 + 4(3) − 10
= 9 + 12 − 10
= 11
4.
2 x 2 − 6 x + 26
Simply each term inside
the parenthesis in order to
arrive with 18 subtracted
by 18 plus 26
= 2 x 2 − 6 x + 26
= 2(3) 2 − 6(3) + 26
= 18 − 18 + 26
= 26
5.
3x 3 − 6
Get the cubed of 3 which is
27, then multiply it to 3 to
get 81 then subtract 6
= 3x 3 − 6
= 3(3) 3 − 6
= 3(27) − 6
= 81 − 6
= 75
Types of Functions
What’s New
Before you proceed to this module, try to look and analyze some of the common types
of functions that you might encounter as you go on with this module.
Types of
Function
Constant
Function
Identity Function
Description
Example
A constant function is a function that has
the same output value no matter what
your input value is. Because of this, a
constant function has the form f ( x) = b ,
where b is a constant (a single value that
does not change).
The identity function is a function which
returns the same value, which was used
as its argument. In other words, the
y=7
5
f (2) = 2
Polynomial
Function
identity function is the function f ( x) = x ,
for all values of x.
A polynomial function is defined by
y = a 0 + a1 x + a 2 x 2 + ... + a n x n , where n is a
0
✓
Linear
Function
✓
Quadratic
Function
✓
Cubic
Function
Power Function
Rational Function
Logarithmic
Function
p( x)
in which numerator, p(x) and
q( x)
denominator, q(x) are polynomial
functions of x, where q(x) ≠ 0.
These are functions of the form:
y = ab x ,
where x is in an exponent and a and b are
constants. (Note that only b is raised to
the power x; not a.) If the base b is greater
than 1 then the result is exponential
growth.
Logarithmic functions are the inverses of
exponential functions, and any
exponential function can be expressed in
logarithmic form. Logarithms are very
useful in permitting us to work with very
large numbers while manipulating
numbers of a much more manageable
size. It is written in the form
y = log b x
Absolute Value
Function
2
A rational function is any function which
can be represented by a rational fraction
say,
Exponential
function
1
non-negative integer and a , a , a
,…, n ∈ R.
The polynomial function with degree one.
It is in the form y = mx + b
If the degree of the polynomial function is
two, then it is a quadratic function. It is
expressed as y = ax 2 + bx + c , where a ≠ 0
and a, b, c are constant and x is a
variable.
A cubic polynomial function is a
polynomial of degree three and can be
denoted by f ( x) = ax 3 + bx 2 + cx + d , where
a ≠ 0 and a, b, c, and d are constant & x
is a variable.
A power function is a function in the form
y = ax b where b is any real constant
number. Many of our parent functions
such as linear functions and quadratic
functions are in fact power functions.
x  0, where b  0 and b  1
The absolute value of any number, c is
represented in the form of |c|. If any
6
y = 2x + 5
y = 3x 2 + 2 x + 5
y = 5 x 3 + 3x 2 + 2 x + 5
f ( x) = 8 x 5
x 2 − 3x + 2
f ( x) =
x2 − 4
y = 2x
y = log 7 49
function f: R→ R is defined by f ( x) = x , it
is known as absolute value function. For
each non-negative value of x, f(x) = x and
for each negative value of x, f(x) = -x, i.e.,
f(x) = {x, if x ≥ 0; – x, if x < 0.
If a function f: R→ R is defined by f(x) =
[x], x ∈ X. It round-off to the real number
to the integer less than the number.
Suppose, the given interval is in the form
of (k, k+1), the value of greatest integer
function is k which is an integer.
Greatest Integer
Function
y = x−4 +2
f ( x) = x + 1
where x is the
greatest integer
function
What is It
Evaluating function is the process of determining the value of the function at the
number assigned to a given variable. Just like in evaluating algebraic expressions,
to evaluate function you just need to a.) replace each letter in the expression with
the assigned value and b.) perform the operations in the expression using the correct
order of operations.
Look at these examples!
Example 1: Given
f ( x) = 2 x − 4 , find the value of the function if x = 3.
Solution:
f (3) = 2(3) − 4
f (3) = 6 − 4
f (3) = 2
Answer: Given
✓
Substitute 3 for x in the function.
✓
Simplify the expression on the right
side of the equation.
f ( x) = 2 x − 4 , f (3) = 2
Example 2: Given g ( x) = 3x 2 + 7 , find
g (−3) .
Solution:
g (−3) = 3(−3) 2 + 7
g (−3) = 3(9) + 7
g (−3) = 27 + 7
g (−3) = 34
7
✓
Substitute -3 for x in the function.
✓
Simplify the expression on the
right side of the equation.
Answer: Given g ( x) = 3x 2 + 7 ,
g (−3) = 34
Example 3: Given p( x) = 3x 2 + 5 x − 2 , find
p(0) and p(−1) .
Solution:
p(0) = 3(0) 2 + 5(0) − 2
p(0) = 3(0) + 0 − 2
p(0) = 0 + 0 − 2
p(0) = −2
Treat each of these like two
separate problems. In each
case, you substitute the value
in for x and simplify. Start with
x = 0, then x=-1.
p(0) = 3(−1) 2 + 5(−1) − 2
p(0) = 3(1) − 5 − 2
p(0) = 3 − 5 − 2
p(0) = −4
Answer: Given p( x) = 3x 2 + 5 x − 2 ,
Example 4: Given
p(0) = −2 , p(−1) = −4
f ( x) = 5x + 1 , find f (h + 1) .
✓
Solution:
✓
f (h + 1) = 5(h + 1) + 1
f (h + 1) = 5h + 5 + 1
f (h + 1) = 5h + 6
Answer: Given
This time, you substitute (h +
1) into the equation for x.
Use the distributive property
on the right side, and then
combine like terms to simplify.
f ( x) = 5x + 1 , f (h + 1) = 5h + 6
Example 5: Given g ( x) = 3x − 2 , find
g (9) .
Solution:
g (9) = 3(9) − 2
✓
Substitute 9 for x in the function.
g (9) = 27 − 2
✓
Simplify the expression on the
right side of the equation.
g (9) = 25
g (9) = 5
Answer: Given g ( x) = 3x − 2 ,
Example 6: Given h( x) =
g (9) = 5
4x + 8
, find the value of function if x = −5
2x − 4
Solution:
8
4(−5) + 8
2(−5) − 4
− 20 + 8
h(−5) =
− 10 − 4
− 12
h(−5) =
− 14
6
h(−5) =
7
h(−5) =
Answer: Given h( x) =
✓
Substitute -5 for x in the function.
✓
Simplify the expression on the right
side of the equation. (recall the
concepts of integers and simplifying
fractions)
4x + 8
6
, h(−5) =
2x − 4
7
Example 7: Evaluate f ( x) = 2 x if x =
3
.
2
Solution:
3
2
3
f =2
2
3
f   = 23
2
3
f = 8
2
3
f   = 4•2
2
3
f =2 2
2
3
for x in the function.
2
✓
Substitute
✓
Simplify the expression on the right
side of the equation. (get the cubed
of 2 which is 8, then simplify)
3
2
Answer: Given f ( x) = 2 x , f   = 2 2
Example 8: Evaluate the function h( x) = x + 2 where x  is the greatest integer
function given x = 2.4 .
Solution:
h(2.4) = 2.4 + 2
✓
Substitute 2.4 for x in the function.
✓
Simplify the expression on the right
side of the equation. (remember that
in greatest integer function, value
was rounded-off to the real number
to the integer less than the number)
h(2.4) = 2 + 2
h(2.4) = 4
Answer: Given h( x) = x + 2 ,
h(2.4) = 4
9
Example 9:Evaluate the function f ( x) = x − 8 where x − 8 means the absolute
value of x − 8 if x = 3 .
Solution:
f (3) = 3 − 8
✓
Substitute 3 for x in the function.
✓
Simplify the expression on the right
side of the equation. (remember that
any number in the absolute value
sign is always positive)
f (3) = − 5
f (3) = 5
Answer: Given f ( x) = x − 8 ,
f (3) = 5
Example 10: Evaluate the function f ( x) = x 2 − 2 x + 2 at
f (2 x − 3) .
Solution:
f (2 x − 3) = (2 x − 3) 2 − 2(2 x − 3) + 2
f (2 x − 3) = (4 x 2 − 12 x + 9) − 4 x + 6 + 2
f (2 x − 3) = 4 x 2 − 12 x + 9 − 4 x + 6 + 2
f (2 x − 3) = 4 x 2 − 12 x − 4 x + 9 + 6 + 2
✓
Substitute 2 x − 3 for x in the
function.
✓
Simplify the expression on the
right side of the equation.
f (2 x − 3) = 4 x 2 − 16 x + 17
What’s More
Your Turn!
Independent Practice 1: Fill Me
Evaluate the following functions by filling up the missing parts of the solution.
1.
f ( x) = 3x − 5 , find f (2)
Solution:
f (2) = ___________________
f (2) = 6 − 5
f (2) = ___________________
2.
g ( x) = 3 2 x , find g(6)
Solution:
10
g (6) = _________________
g (6) = 312
g (6) = _________________
3.
k (a) = a − 2 , find
k (−9)
Solution:
k (−9) = ______________
k (−9) = 9 − 2
k (−9) = ______________
4.
p(a) = −4a − 2 , find p(2a)
Solution:
p(2a) = ______________
p(2a) = ______________
5.
g (t ) = t 2 − 2 , find g (−2)
Solution:
g (−2) = ________________
g (−2) = ________________
g (−2) = ________________
Independent Assessment 1: Evaluate!
Evaluate the following functions. Write your answer and complete solution on
separate paper.
1. Given
w(n) = n − 1, find the value of the function if w = -1.
2. Given f ( x) = x − 3 , find
f (9.3) .
3. Evaluate the function w( x) = − 2 x + 3 if x = -1.
4. Evaluate:
5. Given
f ( x) = − x − 1 , find f (a 2 )
f ( x) = 4 x − 5 , find f (2 x + 3)
Independent Practice 2: TRUE or SOLVE!
11
Analyze the following functions by evaluating its value. Write TRUE of the indicated
answer and solution is correct, if not, rewrite the solution to arrive with the correct
answer on the space provided.
1. Evaluate
f (t ) = 2t − 3 ; f (t 2 )
Solution:
Answer:
f (t 2 ) = 2(t 2 ) − 3
f (t 2 ) = 2t 2 − 3
2. Given the function g ( x) = 5x − 13 , find
g (9) .
Solution:
g (9) = 5(9) − 13
Answer:
g (9) = 45 − 13
g (9) = 32
g (9) = 16 2
3. Given the function f ( x) =
5x − 7
, find the value of the function if x = −3 .
3x − 2
Solution:
5(−3) − 7
3(−3) − 2
− 15 − 7
f (−3) =
−9−2
22
f (−3) =
11
f (−3) = −2
f (−3) =
Answer:
4. Evaluate the function f ( x) = x 2 − 3x + 5 at
f (3x − 1) .
Solution:
f (3x − 1) = (3x − 1) 2 − 3x + 5
f (3x − 1) = 9 x 2 − 6 x + 1 − 3x + 5
f (3x − 1) = 9 x 2 − 9 x + 6
5. Evaluate: g ( x) = 3 x if x =
4
3
Solution:
12
Answer:
4
4
g  = 3 3
3
4
g   = 3 34
3
4
g   = 3 81
3
4
g   = 3 27 • 3
3
4
g   = 33 3
3
Answer:
Independent Assessment 2: Find my Value!
Evaluate the following functions. Write your solution on a separate paper.
1.
g ( x) = 5x − 7 ; g ( x 2 + 1)
Answer: _______________________
2.
h(t ) = x 2 + 2 x + 4 ; h(2)
Answer: _______________________
3. k ( x) =
3x 2 − 1
; k (−3)
2x + 4
Answer: _______________________
4. f ( x) = 2 x 2 + 5 x − 9 ;
f (5x − 2)
Answer: _______________________
5. g ( p) = 4 x ; x =
3
2
Answer: _______________________
What I Have Learned
A. Complete the following statements to show how you understood the different types
of functions. Answer using your own words,
1. A polynomial function is _______________________________________________________
_________________________________________________________________________________.
2. An exponential function _______________________________________________________
_________________________________________________________________________________.
13
3. A rational function ____________________________________________________________
_________________________________________________________________________________.
4. An absolute value function ____________________________________________________
_________________________________________________________________________________.
5. A greatest integer function ____________________________________________________
_________________________________________________________________________________.
B. Fill in the blanks to show how we evaluate functions.
Evaluating function is the process of ___________________________ of the function at
the _________________ assigned to a given variable. Just like in evaluating algebraic
expressions, to evaluate function you just need to ________________________________
in the expression with the assigned value, then _________________________________ in
the expression using the correct order of operations. Don’t forget to
_______________________ your answer.
What I Can Do
In this part of the module, you will apply your knowledge on evaluating functions in
solving real-life situations. Write your complete answer on the given space.
1. Mark charges ₱100.00 for an encoding work. In addition, he charges ₱5.00 per
page of printed output.
a. Find a function f(x) where x represents the number page of printed out.
b. How much will Mark charge for 55-page encoding and printing work?
2. Under certain circumstances, a virus spreads according to the function:
P(t ) =
1
1 + 15(2.1) −0.3t
Where where P(t) is the proportion of the population that has the virus (t) days
after the acquisition of virus started. Find p(4) and p(10), and interpret the results.
14
Assessment
Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a
separate sheet of paper.
1. Which of the following is not a polynomial function?
a.
f ( x) = 2 x − 10
b. g ( x) = 4 x 2 − 3x + 8
c. p( x) = x 3 − 7
d. s ( x) = 3 x − 4 − 9
2. What kind of function is being illustrated by f ( x) =
a.
b.
c.
d.
3x − 11
?
x+7
Rational Function
Constant Function
Greatest Integer Function
Absolute Value Function
3. Find the function value given
h( x) = 9 − 5x of
x = 3m .
a. 9 − 15m
b. 9 − 15m
c. 9 + 15m
2
d. 9 + 15m
4. Which of the following shows an exponential function?
2
a.
f ( x) = 3 x + 8
b. f ( x) = 2 x 3 − 7
c. f ( x) = 3 x − 6
d. f ( x) = x − 8
5. Find the function value given
a.
b.
c.
d.
h( x) = 3x − 8 , if
x = 9a + 1 .
27 a + 5
27 a − 5
18a + 11
18a − 11
6. Which of the following is the value of the function f ( x) = 4 x − 8 + 2 given x = 2?
2
a.
b.
c.
d.
8
9
10
11
15
7. Evaluate the function h( x) = x − 11 given x = 3.5.
a.
b.
c.
d.
-8
8
-9
9
8. Give the value of the of the function c( x) = 3x 2 − 36 at
a.
b.
c.
d.
c(5) .
-21
14
111
39
9. Evaluate: h( x) = 5 x 3 − 3x + 9 given x = 3.
a.
b.
c.
d.
45
63
135
153
10. Find the value of the function
a.
f ( x) = 2 x 2 + 3 if
x = 6.
75
b. 5 3
c.
15
d. 2 3
11. Evaluate the function f ( x) = 2 x 2 − 3x + 1 given x = 3x − 5 .
a. f (3x − 5) = 18 x 2 − 69 x + 66
b. f (3x − 5) = 18 x 2 − 63 x + 51
c. f (3x − 5) = 18 x 2 + 69 x − 66
d. f (3x − 5) = 18 x 2 + 63 x − 51
12. Given g(x) =
x2 − 3
, determine g(5).
2
a. 11
b.
7
2
c. -11
d. −
7
2
13. Evaluate the function g ( x) = 3 x if x =
a.
b.
3
5
.
3
243
243
c. 9 3
d. 33 9
16
14. Given g ( x) =
a.
x 2 − 2x + 5
, determine g (4) .
x+3
5
7
5
7
13
c.
7
13
d. −
7
b. −
15. For what values of x can we not evaluate the function f ( x) =
a.
b.
c.
d.
x+4
?
x2 − 9
±4
±3
±2
±1
Additional Activities
Difference Quotient
f ( x + h) − f ( x)
this quantity is called difference quotient. Specifically, the difference
h
quotient is used in the discussion of the rate of change, a fundamental concept
in calculus.
Example: Find the difference quotient for each of the following function.
A. f(x) = 4x - 2
B. f(x) = x2
Solution:
A. f(x) = 4x - 2
f ( x + h) = 4( x + h) − 2 = 4 x + 4h − 2
f ( x + h) − f ( x) 4 x + 4h − 2 − (4 x − 2)
=
h
h
4 x + 4 h − 2 − 4 x + 2)
=
h
4h
=
h
=4
17
B. f(x) = x2
f ( x + h) = ( x + h) 2 = x 2 + 2hx + h 2
f ( x + h) − f ( x) x 2 + 2hx + h 2 − ( x) 2
=
h
h
2
2
2
x + 2hx + h − ( x)
=
h
2
2hx + h
=
h
= 2x + h
YOUR TURN!
Find the value of
f ( x + h) − f ( x)
, h ≠ 0 for each of the following function.
h
1. f ( x) = 3x + 4
2. g ( x) = x 2 + 3
18
What I
Know
1. A
2. D
3. C
4. B
5. A
6. B
7. D
8. A
9. C
10.B
11.A
12.C
13.D
14.A
15.C
19
What's More
Assessment
Independent Practice 1
1. D
2. A
3. A
4. C
5. B
6. C
7. A
8. D
9. C
10.B
11.A
12.A
13.D
14.C
15.B
1.
2.
3.
4.
5.
Independent Assessment 1
1. -2
2. 6
3. 5
Independent Assessment 2
Independent Practice 2
1. TRUE
4.
5.
2.
3. 2
4.
5. TRUE
1.
2.
3. -13
4.
5. 8
Answer Key
References
Books:
CHED. General Mathematics Learner's Materials. Pasig City: Department of
Education - Bureau of Learning Resources, 2016.
Orines, Fernando B. Next Cantury Mathematics 11. Quezon City: Phoenix
Publishing House, 2016.
Oronce, Orlando A. General Mathematics, 1st Ed. Quezon City: Rex Book Store Inc.,
2016.
Online Sources:
http://www.math.com/school/subject2/lessons/S2U2L3DP.html)
https://www.toppr.com/guides/maths/relations-and-functions/types-offunctions/
20
For inquiries or feedback, please write or call:
Department of Education - Bureau of Learning Resources (DepEd-BLR)
Ground Floor, Bonifacio Bldg., DepEd Complex
Meralco Avenue, Pasig City, Philippines 1600
Telefax: (632) 8634-1072; 8634-1054; 8631-4985
Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph