GENERAL
MATHEMATICS
TERM
:
MODULE CONTENTS
PRELIM
:
LESSON 1
LESSON 2
LESSON 3
LESSON 4
LESSON 5
LESSON 6
LESSON 7
LESSON 8
SUBJECT TEACHER
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:
Relation and functions
Evaluating Function
Fundamental Operation of function – Addition
and Subtraction
Product of Function
Quotients of Functions
Evaluating of composition of function
Rational Equation and inequality Function
Solving rational equation
Summative
MA. KRISTINA B. PABLO, LPT
GENERAL MATHEMATICS
LESSON 1
Relation and functions
Relation – is a set of n ordered pair ( X, Y )
Domain- First coordinate in relation;( X ); input; an independent variable
Range- First coordinate in relation;( X ) ; input; dependent variable
Function- a relation which an element in the domain corresponds to exactly one element of
the range.
Example
Example 2: Not FUNCTION
Example1 1 :
FUNCTION
L
O
1
2
V
3
E
4
TESS
LIEZL
MARIA
JESSICA
BIANCA
JERRY
In Example no. 1 : This is a function
because L is to 1 , O is to 2 , V is to 3 and E
is to 4. Wherein, it shows a one to one
function. Always remember that the
relation is a function when it is a one to one
function. Para bang dapat si x ay meron
lang value na iisa galling kay y.
In Example no. 2 : This is a not function
because Jerry is to Tess, Liezl,Maria, jes and
Bianca. Wherein, it shows a one to many
function. Always remember that the
relation is not a function when it is shows
one to many function. Para bang pag
nangyari na dalawa na kayo sa isang tao
hindi matatawag na function ang relation.
Ways of Representing Function
Mapping
x
3
2
1
4
5
Y
2
4
3
3
-1
Table of values
x
Y
-2
-10
-1
-3
0
4
1
-2
2
1
Graphing
The Vertical Line Test
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If any vertical line passes through a graph at more than one
point, then the graph is not the graph of a
*
function. Otherwise it is the G
graph
E N E of
R AaLfunction.
MATHEMATICS
Mapping : Sa mapping naman para ma identify niyo kung function or not pag in
arrow arrow mo... Gumuhit ng Cartesian plane at I plot doon yung mga ordered
pair at gamitin ang vertical line test . Sa vertical line test malalaman pag ang
dalawang point ay natusok ng line na ilalapat sa cartesian plane ibig sabihin
nun hindi yung function pero pag walang point sa isang line means nun function
yung relation. Itry mo para malaman.
Example for Vertical line test:
A circle is not the graph of a not
as shown below.
function
Types of Relation
1. One to one
X
Y
3
2
2
4
1
3
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Pag nag mahal ka kasi dapat
one to one lang, para mag
FUNCTION! Stick to one dapat
GENERAL MATHEMATICS
2. One to many
X
Y
3
2
2
4
1
3
3. Many to one
X
Y
3
2
2
4
1
3
4. Many to many
x
Sa tunay na relasyon dapat isa lang
baby MO. Hindi yung may baby 1 ,
baby 2 at baby 3 ka pa! kaya walng
nag Function na relationship mo!
Parang si billie naman ito ng meant
to be ; Apat manliligaw .  Tyaka
Ganyan dapat kasi girls! para More
chances of winning. Para mag ing
maganda ang Function ng
relationship mo!
Y
3
2
2
4
1
5
Ikaw nga, san mo ihahalintulad
ito?
__________________________________________
________________________________________.
Activity 1 :
Direction: Use the 3 Ways of Representing Function. Identify if it is a Function or Not
1. ( 3, 2) (-2, 1) (1, 0) (0, 1) (-4 , 3)
2. (0, -3) (-4, 0 ) (3, -4) (1, 4) (-5, 8)
3. (1, 4) (2, -4) (3, 1) (-3, -4) (-4, 2)
4. (-1, 3) (-4, -4) (-2, -1) (3, -4) (4, -2)
5. (0, 3) (-2, 1 ) (-5, 4) (-2, 4) (-5, -3)
Activitivity 1.a
Capture things from your home that will illustrate the rule of a function
LESSON 2
Evaluating Function
To evaluate a function is to:
Replace (Substitute) its variable with a given number or expression.
Like in this example:
Example: evaluate the function f(x) = 2x+4 for x=5
Just replace the variable "x" with "5":
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GENERAL MATHEMATICS
f(5) = 2×5 + 4 = 14
Answer: f(5) = 14
Note : F is the name of a function; it could be (g, h)
2x + 4 is the variable of the function
Example : Follow the PEMDAS
1. X = 3 , if f(x) = 5x+3 ,
P-ARENTHESIS
f(3) = 5(3) + 3
f(x) = 18
E-XPONENT
2. X = 2 , If f(x) = x2 + 5x -3
f(2) = 22 + 5(2) -3
M-ULTIPLICATION
= 4 + 10 – 3
D-IVISION
f(x) = 11
3. X = -2 , If
f(x) = x2 + 5x -3
A-ADDITION
2
f(2) = -2 + 5(-2) -3
S-UBTRACTION
= 4 + (-10 ) – 3
f(x) = -9
Activity 2:
Evaluate the Following if x= 2 and -3 Show your Solution.
1. 2x3+ 4x2 – 3x + 8
2. X + 23
3. -3x2 + 2x -6
4. 5x3+ 8x2 – 2x + 10
5. 2x2 – 8x +18
Figure 1. Types of Function
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GENERAL MATHEMATICS
LESSON 3
Fundamental Operation of function
Operations with Functions
We can add, subtract, multiply and divide functions!
The result is a new function.
Let us try doing those operations on f(x) and g(x):
Addition
Steps :
1. Substitute the value of the given functions
2. Combine like terms’
We can add two functions:
(f+g)(x) = f(x) + g(x)
Note: we put the f+g inside () to show they both work on x.
Example: f(x) = 2x+3 and g(x) = x2
(f+g)(x) = (2x+3) + (x2) = x2+2x+3
Sometimes we may need to combine like terms:
Example 2
Problem
Find (f + g)(x).
Identify f(x) and g(x).
Replace f(x)
with
, and g(x)
with
.
Then add and combine
like terms.
Answer
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GENERAL MATHEMATICS
Acitivity 3.a
Operation of a Function : Addition
1. f (x) = (2x +3 )
g (x) = (3x -2)
Note : lagi mong tatandaan pag sa addition g
function, ang kailangan mong gawin pag
sama samahin yung mga parehas ng variables
(xyz) then just add or kung nalilito ka. Pag
tapat-tapatin mo. Then add the coefficient.
2. f (x) = (4x3 + 2x2 – 3x + 12 )
g (x) = (-2x3 + 2x2 + 12 )
3. f (x) = ( 3x2+ 2x -3x )
g (x) (2x +3 )
4. f (x) = 4x2 + 2x -10
g (x) = x2 –x – 3
5. (3x - 3 ) (2 x -4 )
Yung variables, yun yung mga letters while
yung coefficient e yung numbers.
Subtraction
Steps
1. Substitute the value of the given functions
2. Change the sign of the Subtrahend
3. Combine like terms
We can also subtract two functions:
(f-g)(x) = f(x) − g(x)
Example: f(x) = 2x+3 and g(x) =-x+2
(f-g)(x) = (2x+3) − (-x+2)
(f-g)(x) = (2x+3) + (x-2)
(f-g)(x) = 3x +1
Note : lagi mong tandaan ito KCC
KEEP the minuend yung nauna
CHANGE the sign
CHANGE the subtrahend yung
pangalawa. Paitan yung mga sign like positive
to negative o vice vers
Find (g – f)(x).
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GENERAL MATHEMATICS
Replace g(x) and f(x) with
their respective expressions.
Then subtract and combine
like terms.
Answer
Activity 3.b
Operation of a Function: Subtraction
1. f (x) = (2x +3 )
g (x) = (3x -2)
2. f (x) = (4x3 + 2x2 – 3x + 12 )
g (x) = (-2x3 + 2x2 + 12 )
3. f (x) = ( 3x2+ 2x -3x )
g (x) (2x +3 )
4. f (x) = 4x2 + 2x -10
g (x) = x2 –x – 3
5. (3x - 3 ) (2 x -4 )
LESSON 4
Product of Function
Multiplication
1. Product rule for Exponent
Xm ∙ Xn = Xm + n, this says that to multiply two exponents with the same base, you keep
the base and add the powers.
2. Power rule for Exponent
(Xm)n = Xmn, this says that to raise a power to a power you need to multiply
the exponents. There are several other rules that go along with the power rule, such as
the product-to-powers rule and the quotient-to-powers rule.
3. Power of a product Rules
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GENERAL MATHEMATICS
The Power of a Product rule states that a term raised to a power is equal to
the product of its factors raised to the same power. In this lesson, learn more about
this rule
(xy)n =xnyn
Polynomials : comes from poly- (meaning "many") and -nomial (in this case meaning "term")
... so it says "many terms"
What is Monomial?
An algebraic expression consisting of one term.
What is Binomial?
An algebraic expression consisting of two terms.
What is Trinomial?
An algebraic expression consisting of three terms.
There is also quadrinomial (4 terms) and quintinomial (5 terms),
but those names are not often used.
Use distributive property: yung isang term
a. Multiplying Binomial to monomial
I multiply mo sa lahat ng numbers sa
f = 3x g = 2x +3
isang set of number. After nun I arrange
(f∙g)(x) = ( 3x ) (2x +3)
mo. From highest degree to lowest,
(f∙g)(x) = = 6x2 +9x
means yung exponent ang pinag
uusapan pag degree
Multiply
by
2x
3x
6x2
3
9x
3x times 2x is equal to 6x2 ilagay mo sa
box katapat nila.
Mam , bakit x2 yung sa 6? Dahil ang
exponent sa multiplication ay pinag
aadd. Nandun sa taas yung rule na
binasa mo.
then 3x times 3 is equal to 9x
pag natapos nay an, ayusin mo na.
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GENERAL MATHEMATICS
b. Multiplying binomial to binomial
f = 2x + 2
g = 5x -4
(f∙g)(x) = (2x + 2 ) (5x -4)
Last
Inner
(f∙g)(x) = (2x + 2 ) (5x -4)
First
Multiply
by
2x
2
Outer
Laging tatandaan ang FOIL method ay
magagamit lang sa binomial to
binomial. First means yung unang
terms ng both set of equation, next
yung Outer I multiply si first term ng
unang set sa last ng second equation.
Inner naman yung dalawang loob.
Last, yung dulu duluhan ng dalawang
equation.
= (2x∙5x) (2x∙-4) (2∙5x) (2∙-4)
= (10x2) (-8x) (10x) (-8)
= 10x2 + 2x -8 Combine like terms
5x
-4
Ang Box method ay applicable sa
lahat. Mam, bakit may bilog yung -8x
10x2
-8x
at 10x? kasi same sila ng variable ibig
sabihin dapat I simplify sila.
10x
-8
FOIL Method – this is only applicable for Binomial to Binomial
c. Multiplying binomial to Trinomial
f = -3x + 5 g = 5x2 -2x + 4
(f∙g)(x) = (-3x + 5) (5x2 -2x + 4)
=(-3x ∙ 5x2) (-3x ∙(-2x)) (-3x ∙ 4) (5 ∙ 5x2 ) (5 ∙(-2x) ) (5 ∙ 4)
= ( -15 x3) (6x2)(-12x) (25x2)(-10x)(20)
= ( -15 x3) (6x2) (25x2) (-12x)(- 10x) (20)
Combine like terms = -15 x3 + 31 x2 -22x + 20
Multiply by
5x2
-2
+4
-3x
-15 x3
6x2
-12x
5
25x2
-10
20
Subukan niyo to, para masanay
at ma master ninyo. Same
process lang to sa naunang mga
examples
Combine like terms = -15 x3 + 31 x2 -22x + 20
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GENERAL MATHEMATICS
Example
Problem
Find the product of f and g.
To find the product,
multiply the functions.
Replace f(x) with (2x + 1),
and g(x) with (5x – 3).
Answer
Activity 4
Operation of a Function: Multiplication
1. f (x) = (2x +3 )
g (x) = (3x -2)
2. f (x) = (4x3 + 2x2 – 3x + 12 )
g (x) = (-2x3 + 2x2 + 12 )
3. f (x) = ( 3x2+ 2x -3x )
g (x) (2x +3 )
4. f (x) = 4x2 + 2x -10
g (x) = x2 –x – 3
5. (3x - 3 ) (2 x -4 )
LESSON 5
Quotient of Functions
Quotient Rule of Exponent
1.
2.
3.
𝑥𝑚,
𝑥𝑛
𝑥𝑚,
𝑥𝑛
𝑥𝑚,
𝑥𝑛
= 𝑥 𝑚−𝑛 𝑤ℎ𝑒𝑛 𝑚 > 𝑛
1,
= 𝑥 𝑛−𝑚 𝑤ℎ𝑒𝑛 𝑚 < 𝑛
= 𝑥 0 𝑤ℎ𝑒𝑛 𝑚 = 0
𝑥3,
𝑥3
𝑥5,
𝑥2
𝑥4,
𝑥6
=𝑥 5−2 = 𝑥 3
1,
=𝑥 6−4 =
1,
𝑥2
= 𝑥3 = 1
Division
Example:
1. f (x ) = 10 x 20
𝑓(𝑥),
𝑔(𝑥)
=
10𝑥 20 ,
2𝑥 8
= 5𝑥12
2. f (x ) = = -8 x 5
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g (x ) = 2 x 8
g (x ) = 4 x 8
GENERAL MATHEMATICS
𝑓(𝑥),
𝑔(𝑥)
=
−8𝑥 5 ,
4𝑥 8
3. f (x ) = = 6x 3
𝑓(𝑥),
𝑔(𝑥)
=
6𝑥 3 ,
6𝑥 3 ,
1
= −2 (𝑥 5−8 ) =
−2
𝑥3
g (x ) = 6 x 3
= 1(𝑥 0 )
Example
Problem
Find
.
To find the quotient,
divide f by g.
Substitute the polynomials in
for f(x) and g(x) and divide.
We add
because x = 0
would make the
denominator g(x) =0
and
undefined.
Remember to rename
1.
as
Answer
Activity 5
Operation of a Function: Division
1. f (x ) = 36x 24
g (x ) = 18 x 12
2. f (x ) = 4x 5
g (x ) = 4 x5
3. f (x ) = 12 x 3
g (x ) = 2 x 14
4. f (x ) = 6 x 6
g (x ) = 2 x 22
5. f (x ) = x – 5
g (x ) = 10x - 20
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I apply ang rules of quotient.
E.1 : 10 divide by 2 is equal to 5,
then 20 minus 8 is equal to 12 so
the answer is 5x raise to 12. Apply
the QRE#1
2. Apply the QRE#2
3. Apply the QRE#3
GENERAL MATHEMATICS
LESSON 6
Evaluating of composition of function
(f°g)(x)
First
Second
Steps
1.
2.
3.
4.
5.
6.
Given
Replace the “x” by the second function
Distribute
Combine similar terms
Substiture the value of x
Solve and Simplify
Example :
1. f = 4x +3 , g = x2 + 3
f°g
=4 x2 + 15
g°f
= x2 + 3
=(4x +3) 2 + 3
=16x2 + 24x +12
=4(x2 + 3) + 3
=4 x2 + 12 + 3
2. f = -20x , g = x2 -40
f°g
=-20(x2 -40)
=-20 x2 + 800
g°f
= -202 -40
= 400 -40
= 360
LESSON 7
Rational Equation and inequality Function
Polynomial is an expression consisting of variables (also called indeterminate
) and coefficients, that involves only the operations of addition, subtraction,
multiplication, and non-negative integer exponentiation of variables
Example :
1. 2x + 3 : P
4. 5x-4 :NP
2. 2y3 – 4y -10 : P
5. 45x2 – 12 x +3 : P
3. √481 : NP
Rational Equation: an equation involving rational expressions
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GENERAL MATHEMATICS
Rational Inequality: Inequality involving rational expressions
Rational Function: A function that is the ratio of two polynomials.
It is "Rational" because one is divided by the other, like a ratio.
(Note: the polynomial we divide by cannot be zero.)
LESSON 8
Solving rational equation
Steps : IDEMSOF
1.
2.
3.
4.
Identify the LCD of ll the denominator
Multiply each term of the equation by the LCD
Solve the resulting equation
Finalize your answer
Example 1.
4𝑥 1 1
+ =
6 3 2
IDE : 6 (LCD)
4𝑥 1 1 6
( + = )
6 6 2 1
M
24𝑥
(
6
6
+ = )
3
2
6
Activity:
x=1/4
2
4
6
3
3
12
−
12
=1
1
2.
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4
F
− =
6
4𝑋
6
2𝑥
So
1.
3
4𝑥
4x + 2 =3
4X = 3 – 2
=
4
GENERAL MATHEMATICS
SUMMATIVE:
EVALUATE THE FOLLOWING: x = 4 , y= -2 , and x = -8, y = 6 , z = -5
1. 8x2y2+ 2xy – 8
2. 6xyz + 3
3. -4x4y + 12 x 2 – 3x + 2
4. X2 + 4xy + 2
5. X3 + 2x2y –x + 4
Plot the following ordered pairs and Identify if it is Function or not. VERTICAL LINE TEST
AND MAPPING
1. ( -3, 4) (2, -1) (-1, 0) (3, -1) (4 , -3)
2. (-2, 4) (-2, 4) (2,1) (-2, -5) (-4, 3)
3. (-3, 2) (2, 1) (1, 0) (0, 1) (-4 , -3)
4. (3, -3) (2, -4) (-2, -1) (5, -3) (1, -2)
5. ( -2, 2) (-4, 1) (-1, 3) (0, 3) (-2 , 3)
Operation of Function (addition, subtraction, multiplication for 1-4)
1. (3x - 3 ) (2 x -4 )
2. (-2x + 2) ( 2x )
3. (6xy + 3) (-8X-2)
4. (4x4 – 6x 3 + 3x 2 -3x +12) (x2 +4x -3)
5. Find (f+g/h)(x)
F(x) = 8x3-3x2
g(x) = 4x3 + 9x2
h(x) = 3x2
Composition of Function
F = x2 +4x -3
G = 4x4 – 6x 3 + 3x 2 -3x +12
H = 12x -4
a. ( F o g ) (x)
b. ( g o f ) (x)
c. ( h o f ) (x)
d. ( g o h ) (x)
e. ( F o g o h ) (x)
God bless!
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GENERAL MATHEMATICS