Page 17 - Mathematics_Form_3

Relations

Solution relation R is {y : y is a set of all
(a) The relation R = {(x,y): y ≤ x} has real numbers} .
an inequality sign ʻ ≤̕ which means (b) The relation R = {(x , y ) : y < x} has
that all the points on the line y = x an inequality sign < which means
are included in the required region. that all the points on the line y = x
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In this case, a solid line is used. are not included in the required
Since the line divides the xy-plane
into two regions, test at least one region. Therefore, in this case, a
point from each region to determine dotted line is used. Since the dotted
the region which satisfies the line separates the xy-plane into two
relation. For instance, from upper regions, test at least one point from
half region, choose (–1, 4) and each region. For instance, choose
(–3, 2) and from the lower region, (– 4, 4), and (– 2, 0) from the upper
choose (2, 1) and (4, 2). Substituting half of the region and (4, – 4) and
the values of the ordered pairs (1, – 1) from the lower half of the
of x and y into the inequality region. Observe that (1, –1) and
gives 2<–3 and 4<–1 , and 1<2 (4, – 4) satisfy the inequality, while
and 2<4 , respectively. Hence, (2, 1) (– 4, 4) and (–2, 0) do not satisfy.
and (4, 2) satisfy the inequality, Shade the lower region as it contain
while (–3, 2) and (–1, 4) do not points which satisfy relation R as
satisfy the inequality. Shade shown in the following figure.
the lower region which contain
points which satisfy the relation (-4, 4) 4 y
R = {(x,y): y ≤ x} as shown in the 3
following figure. y=x
2
y 4 3 y=x -4 -3 -2 -1 0 1 (1,-1) 3 4 5 x
1
(-1, 4)
Mathematics for Secondary Schools -3 -2 -1 0 1 (2, 1) 3 (4, 2) From the graph, the domain and
(-2, 0)
2
-1
(-3, 2)
2
-2
1
-3
-4
(4,-4)
4 x
2
-1
-2
-3
real numbers as all values of x and
From the graph, the domain of the range of the relation R is a set of all
y are involved, since the line y = x
relation R is {x : x is a set of all real extends infinitely in both directions.
numbers} and the range of the
(c) The relation R = {( , ):xy y >− 2x + } 1 .

10 Student\s Book Form Three

18/09/2025 09:58:38
MATHEMATIC F3 SB.indd 10
MATHEMATIC F3 SB.indd 10 18/09/2025 09:58:38