Islamic republic of Afghanistan
Ministry of Education
Afghan Turk Maarif high schools
Mathematic Lessons for
th
10 Class
lecturer: Shekib Moushfiq
year: 2020
Chapter 2
Two ordered pairs are equal if and only if corresponding
components are equal to each other.
(a, b) = (c, d) if a=c and b=d
Ex: Given that (2x – 1, 7) = (3, 3y-8), find x and y
Ex: Given that (6, 2x-y) = (x+y, -3), find x and y
Find the unknowns in the following ordered pairs
a. (3x + 5, 4) = (2, y)
b. (x – 2y, 3x + y) = (2, 3)
Ex: Represent the relation having ordered pairs of the form (x, y)
such that 𝒚𝟐 = x in the domain {0, 1, 4} by list and graph.
Find points:
a. A (3, 4)
b. B (2, -3)
c. C (-4, 2)
d. D (-3, -4)
In the coordinate plane.
If A and B are two sets. The set of all ordered pairs,
whose first component is from A and second
component is from B, Is called the Cartesian product of
A and B and it is denoted by A × 𝐁.
If set A = {1, 2, 3} and B = {x, y}
Find:
a. A × 𝑩
a. B × 𝑨
Given two sets A and B
n (A × 𝑩) = n (B × 𝑨) = n (𝑨) × n (B)
Ex: If set A = {1, 2, 3} and B = {x, y}
Find:
n (A × 𝑩) =?
1. Given that A = {1, 2, 3} and B = {a, b, c, d}
Find:
a. A × 𝑩
b. B × 𝑨
c. n (A × 𝑩)
2. Plot the pairs (2, 0), (3, -3), (-4, -5) in the coordinate plane.
Definition: a relation is a set of ordered pairs.
The set of the first component is called domain and the set of
all second component is called the range of the relation.
Let's say {(0, 2), (1, -5), (2, 6), (6, -7)} is a relation. Find its
domain and range.
Domain: {0, 1, 2, 6}
Range: {2, -5, 6, -7}
The set of domain and range of a relation is also denoted
by mapping.
1. Represent the relation {(Monday, sunny), (Tuesday, rainy), (Wednesday,
sunny), (Thursday, cloudy), (Friday, cloudy)} by mapping and graph.
2. Represent the relation having ordered pairs of the form (x, y) such that
y2 = x in the domain {0, 1, 4} by list and graph.
3. Given the domain [–3, 4] and the relation containing ordered pairs of the
form (x, y) such that y = –2x + 3 represent the relation by mapping and
find its range.
If we change the place of domain and range in a relation set
so we achieve the inverse of relation:
Ex: {(0, 2), (1, -5), (2, 6), (6, -7)} is a relation find inverse of this
relation?
You can easily guess the rule that relates the number on the left to
the number on the right. X 𝑥 2 and we can write
f(x) = 𝒙𝟐
Definition: A function is a relation in which the first component is
associated with exactly one second component.
Or: a function f is a rule that assign to each element x in set A
exactly one element y or f(x) in set B.
Set A is called domain and set B is called the range of the function f.
Ex: State whether the following relations define a function
or not?
a. {(0, 2), (0, 3), (1, 6), (2, 4), (3, 5)}
b. {(–3, 1), (–1, –1), (0, 1), (1, 3), (2, –2)}
A function can also be thought of as a set of ordered pairs
whose first components are all different.
Ponder:
In order to have a function for each value in the domain we
should have exactly one element assigned in the range.
For defining a function always keep mind in this example:
This is a
function
because
each son
has only
one mother.
Ex: Determine whether the following relations are function or
not?
Ex: Determine whether the following relations are function or
not?
Example:
Given the function f(x) = 𝒙𝟐 − 𝟑𝐱
Find:
a. f(4) = ?
b. F(-6) = ?
c. F(10) = ?
Example:
1. Given the function f(x) = 𝑥 2 – 5x, find f(0), f(3), f(–a), f(x + 1).
3. 0, –6, a2 + 5a, x2 – 3x – 4 4. 4, 4, 20
Graph of a Function
The graph of a function f is the collection of ordered pairs (x, f(x)) such
that x is in the domain of the function.
Ex: Sketch graph of f(x) = x – 2 if D( f ) = {–2, –1, 0, 1, 2, 3}.
Ex: Sketch graph of f(x) = x – 2.
X-Intercepts of a Function
If the graph of a function, f, crosses the x-axis, then the function has an
x-intercept. Generally, an x-intercept is written as the ordered pair (a, 0)
where a is any real number.
To find x-intercepts, we need to find all x-values that satisfy the equation
f(x) = 0. Since an equation may have more than one solutions, graph of a
function can cross x-axis more than once.
Ex: Find the zeros or x-intercepts of the following functions:
1. f(x) = 2x + 10
2. g(x) = 𝒙𝟐 – 9
Y-Intercepts of a Function
Similarly, if the graph of a function, f, crosses the y-axis, then the graph of
the function has a y-intercept. Generally, the y-intercept is written as the
ordered pair (0, b) where b is any real number.
To find y-intercepts, you need to let x = 0, and then solve for y. Since we
are talking about a function, f(0) can have only one value. That means
graph of a function can cross y-axis only once.
Note: A function can have only one y-intercept.
Ex: Find the zeros or y-intercepts of the following functions:
1. f(x) = 2x + 10
2. g(x) = 𝒙𝟐 – 9
𝒙+𝟑
Ex: Find the y-intercept of the function f(x) =
𝒙𝟐 +𝟏
Vertical Line Test for Functions
By definition, a function needs to have at most one y-value assigned to
each x-value. That is, a vertical line can cross the graph of the function
at most once for any x-value.
A set of points in the coordinate plane is the graph of a function if and
only if no vertical line crosses the graph at more than one point.
Ex: Which of the following graphs are functions? Why or why not?
Ex: Which of the following graphs are functions? Why or why not?
Ex: Sketch any function whose x-intercepts are –3, 2, 4 and whose
y-intercept is 3 so that the graph passes through the point (1, 1.5).
Home Work
1. Sketch graph of the function f(x) = 2x + 4
2. Find the x-intercept(s) of the function f(x) = 3x – 1
3. Find the y-intercept of the function g(x) = x2 – 2x + 7.
4. State whether the following are functions or not. Explain your answer.
5. Sketch graph of any function with x-intercepts –2, 0 and y-intercept 0 so
that graph passes through the point (3, 1).
