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General Education 4: Mathematics in the Modern World
First Semester Academic Year: 2020 2021 | 1
Function and Relation
Relation is a correspondence between two things or quantities.
Domain it is a set of ordered pairs such that the set of all first coordinates of the ordered pars.
Range the set of all the second coordinates of the ordered pairs.
Example:
R = {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}
Type of elation
1. One to one relation
2. One to many relation
3. Many to one relation
Function is a relation such that each element of the domain is paired with exactly one element
of the range. A function is a relation in which no two ordered pairs have the same first
coordinate. We use the function notation as:
y = f(x)
where : f = indicate the function exists between variables x and y.
Evaluating a Function
The functional notation y = f(x) allows us to denote specific values of a function. To
evaluate a function is to substitute the specified values of the independent variable in the formula
and simplify.
Example1:
When f(x) = 2x 3, find the f(2)
Solution:
f(2) = 2(2) 3
= 4 3
= 1
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General Education 4: Mathematics in the Modern World
First Semester Academic Year: 2020 2021 | 2
Example 2:
If f(x) = 2x
2
3x +5; find a. f(4) b. f(-3) c. f(5)
Solution:
a. f(4) = 2(4)
2
3(4) + 5 b. f(-3) = 2(-3)
2
3(-3) + 5
= 2(16) 12 + 5 = 2(9) (-9) + 5
= 32 12 + 5 = 18 + 9 + 5
= 20 + 5 = 32
= 25
c. f(5) = 2(5)
2
3(5) + 5
= 2(25) 15 + 5
= 50 15 + 5
= 35 + 5
= 40
Fundamental Operation on Functions
1. The sum of two function f and g is the functions defined by (f±g)
(
x
)
= 𝑓
(
𝑥
)
± g(x).
2. The product of two functions f and g is the function defined by (fg)(x) = f(x) g(x)
3. The quotient of two function f and g is defined by the function (f/g)(x) = f(x)/g(x), g(x) ≠ 0
Example:
If f(x) = 8 3x and g(x) = 5 x, find a. (f+g) (x) b. (f-g) (x) c. (fg)(x)
d. (f/g)(x)
Solution:
a. (f+g)(x) = f(x) + g(x) c. (fg)(x) = f(x) g(x)
= (8 3x) + (5 x) = (8 3x) (5 x)
= 8 +5 3x x = 40 8x -15x +3x
2
= 13 4x = 40 23x + 3x
2
b. (f-g)(x) = f(x) - g(x) d. (f/g)(x) = f(x) /g(x)
= (8 3x) + (5 x) = (8 3x) / (5 x)
= 8 -5 3x x
= 3 2x
Operations
A binary operation on a set A is a function that takes pairs of elements of A and produces
elements of A from them.
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General Education 4: Mathematics in the Modern World
First Semester Academic Year: 2020 2021 | 3
Properties of Operation.
1. Commutative x*y = y*x
2. Associative x*(y*z) = (x*y) * z
3. Identity e*x = x * e
4. Inverse x * y = y * x = e
Exercise 2.1
For each of the following functions, using separate sheets find the following:
a. A general formula for f + g , f g, and f/g.
b. The value of each combination at x = 3 and x = -2
1. f(x) = 4x -5 and g(x) = 5 2x
2. f(x) = 7 - 3x and g(x) = x
2
3x
3. f(x) = 9 –x2 and g(x) = √5 – 2x
4. f(x) = 2/x and g(x) = x
2
+ 3x + 2
5. f(x) = 3x -4 and g(x) = √x
2
4x +3
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General Education 4: Mathematics in the Modern World
First Semester Academic Year: 2020 2021 | 4
Lesson 2.2: Mathematical as a Language
Mathematical language, there is also an expression (name given to mathematical object
of interest) and a mathematical Sentence (just like the English language must state a complete
thought).
Example of Mathematical Expression are the following:
1. numbers
2. set
3. matrix
4. ordered pair
5. average
Proper writing of mathematical sentences aids to the proper solving of problems and
proofs of theorems or conjectures.
Expression
Mathematical Expression
A number increased by seven.
x + 7
Thrice a number added to ten.
3x + 10
One number is four times the other
x, 4x
Sum of three consecutive integers
x + (x + 1) + (x + 2)
Ten less than four times a certain number
4x 10
Mother is six years more than three times older
her son.
If x is son’s age, then three times son’s age 3x
and six more than 3x is 3x + 6

Unformatted Attachment Preview

General Education 4: Mathematics in the Modern World Function and Relation Relation –is a correspondence between two things or quantities. Domain – it is a set of ordered pairs such that the set of all first coordinates of the ordered pars. Range – the set of all the second coordinates of the ordered pairs. Example: R = {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)} Type of elation 1. One to one relation 2. One to many relation 3. Many to one relation Function – is a relation such that each element of the domain is paired with exactly one element of the range. A function is a relation in which no two ordered pairs have the same first coordinate. We use the function notation as: y = f(x) where : f = indicate the function exists between variables x and y. Evaluating a Function The functional notation y = f(x) allows us to denote specific values of a function. To evaluate a function is to substitute the specified values of the independent variable in the formula and simplify. Example1: When f(x) = 2x – 3, find the f(2) Solution: f(2) = 2(2) – 3 =4–3 =1 First Semester Academic Year: 2020 – 2021 |1 General Education 4: Mathematics in the Modern World Example 2: If f(x) = 2x2 – 3x +5; find a. f(4) b. f(-3) c. f(5) Solution: a. f(4) = 2(4)2 – 3(4) + 5 = 2(16) – 12 + 5 = 32 – 12 + 5 = 20 + 5 = 25 b. f(-3) = 2(-3)2 – 3(-3) + 5 = 2(9) – (-9) + 5 = 18 + 9 + 5 = 32 c. f(5) = 2(5)2 – 3(5) + 5 = 2(25) – 15 + 5 = 50 – 15 + 5 = 35 + 5 = 40 Fundamental Operation on Functions 1. The sum of two function f and g is the functions defined by (f±g)(x) = 𝑓(𝑥) ± g(x). 2. The product of two functions f and g is the function defined by (fg)(x) = f(x) g(x) 3. The quotient of two function f and g is defined by the function (f/g)(x) = f(x)/g(x), g(x) ≠ 0 Example: If f(x) = 8 – 3x and g(x) = 5 – x, find a. (f+g) (x) d. (f/g)(x) b. (f-g) (x) c. (fg)(x) Solution: a. (f+g)(x) = f(x) + g(x) = (8 – 3x) + (5 – x) = 8 +5 – 3x – x = 13 – 4x c. (fg)(x) = f(x) g(x) = (8 – 3x) (5 – x) = 40 – 8x -15x +3x2 = 40 – 23x + 3x2 b. (f-g)(x) = f(x) - g(x) = (8 – 3x) + (5 – x) = 8 -5 – 3x – x = 3 – 2x d. (f/g)(x) = f(x) /g(x) = (8 – 3x) / (5 – x) Operations A binary operation on a set A is a function that takes pairs of elements of A and produces elements of A from them. First Semester Academic Year: 2020 – 2021 |2 General Education 4: Mathematics in the Modern World Properties of Operation. 1. 2. 3. 4. Commutative Associative Identity Inverse x*y = y*x x*(y*z) = (x*y) * z e*x = x * e x*y=y*x=e Exercise 2.1 For each of the following functions, using separate sheets find the following: a. A general formula for f + g , f – g, and f/g. b. The value of each combination at x = 3 and x = -2 1. 2. 3. 4. 5. f(x) = 4x -5 and g(x) = 5 – 2x f(x) = 7 - 3x and g(x) = x2 – 3x f(x) = 9 –x2 and g(x) = √5 – 2x f(x) = 2/x and g(x) = x2 + 3x + 2 f(x) = 3x -4 and g(x) = √x2 – 4x +3 First Semester Academic Year: 2020 – 2021 |3 General Education 4: Mathematics in the Modern World Lesson 2.2: Mathematical as a Language Mathematical language, there is also an expression (name given to mathematical object of interest) and a mathematical Sentence (just like the English language must state a complete thought). Example of Mathematical Expression are the following: 1. 2. 3. 4. 5. numbers set matrix ordered pair average Proper writing of mathematical sentences aids to the proper solving of problems and proofs of theorems or conjectures. Expression A number increased by seven. Thrice a number added to ten. One number is four times the other Sum of three consecutive integers Ten less than four times a certain number Mother is six years more than three times older her son. First Semester Mathematical Expression x+7 3x + 10 x, 4x x + (x + 1) + (x + 2) 4x – 10 If x is son’s age, then three times son’s age 3x and six more than 3x is 3x + 6 Academic Year: 2020 – 2021 |4 Name: Description: ...
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