3. A relation is a set of
ordered pairs (x,y).
RELATIONS
A relation is a rule that relates
values from a set of values (called
the domain) to a second set of
values (called the range).
A function is a set of ordered pairs
(x,y) such that no two ordered
pairs have the same x-value but
different y-values
FUNCTIONS
A function is a relation where each
element in the domain is related to
only one value in the range by some
rule.
DEFINITIONS
SET OF ODERED PAIR (X,Y)
X – First Element (Independent variable) , - DOMAIN
Y – Second Element (Dependent variable), - RANGE
4. Example
1.Which of the following relations are
functions?
a) f ={(1,2),(2,2),(3,5),(4,5)}
b) g ={(1,3),(1,4),(2,5),(2,6),(3,7)}
c) h = {(1,3),(2,6),(3,9),…,(n,3n)}
SOLUTION
The relations f and h are functions
because no two ordered pairs have the
same x-values but different y values,
while g is not a function because (1,3)
and (1,4) are ordered pairs with the
same x-value but different y-values.
Relations and functions can be represented by mapping diagrams where
the elements of the domain are mapped to the elements of the range
using arrows.
5. Relations and functions can be represented by mapping diagrams where
the elements of the domain are mapped to the elements of the range
using arrows.
SET OF ORDERED PAIR
A.) {(3,2), (4,0), (5,1), (2,3)} FUNCTION
B.) {(1,2), (0,3), (1,6), (5,4)} NOT FUNCTION
C.) {(3,4), (3,0), (3,1), (3,3)} NOT FUNCTION
D.) {(4,2), (3,2), (6,2), (5,20} FUNCTION
6. EXAMPLE 2
Which of the following mapping diagrams represent functions?
Solution. The relations f and g are functions because each value y in Y is unique for a
specific value of x. The relation h is not a function because there is at least one element in
X for which there is more than one corresponding y-value. For example, x=7 corresponds to
y = 11 or 13. Similarly, x=2 corresponds to both y=17 or 19
7. A relation between two sets of
numbers can be illustrated by
a graph in the Cartesian
plane, and that a function
passes the vertical line test.
THE VERTICAL LINE TEST
A graph represents a function if and only if
each vertical line intersects the graph at
most once.
11. Evaluating a function means replacing the variable in the
function, in this case x, with a value from the function's domain
and computing for the result. To denote that we are evaluating
f at x for some x in the domain of f, we write f(x),
Example 1:
find f(x) where x = 1.5
a. f(x)=2x+1
b. f(x)=x^2-2x+2
c. f(x)=√(x+1)
d. f(x)=(2x+1)/(x-1)
Solutions
a. f(1.5)=2(1.5)+1=4
b. f(1.5)=(1.5)^2-2(1.5)+2=2.25-3+2=1.25
c. f(1.5)=√((1.5)+1)=√2.5
d. f(1.5)=(2(1.5)+1)/((1.5)-1)=(3+1)/0.5=8
13. Definition. Let f and g be functions
Their sum, denoted by f + g, is the function denoted by
(f+g)(x)=f(x)+g(x)
Their difference, denoted by f - g, is the function denoted by
(f-g)(x)=f(x)-g(x)
Their product, denoted by f . g, is the function denoted by
(f.g)(x)=f(x).g(x)
Their quotient, denoted by f / g, is the function denoted by
(f/g)(x)=f(x)/g(x), excluding the values of x where g(x)=0
14. Use the following functions
below
a. f(x)=x+3
b. p(x)=2x-7
c. v(x)=x^2+5x+4
d. g(x)=x^2+2x-8
e. h(x)=(x+7)/(2-x)
f. t(x)=(x-2)/(x+3)
Example 5
Determine the following
functions.
a. (v+g)(x)
b. (f.p)(x)
c. (p-f)(x)
17. Definition. Composite functions are functions that involve
substitution of functions, such as f(x) is substituted for the x-
value in the g(x) function or the reverse. Which goes where is
outlined by the way the equation is written:
The more conventional way to write these composite functions
is:
19. Name:
Set: Contact No.
Supplementary Exercises
1. Which of the following statements represents a function?
a. Student to their current age.
b.Countries to its capital
c. A store to its merchandise
2. Which of the following letters will pass the vertical line test? V W X Y Z
3. A person is earning P600 per day to do a certain job. Express the total salary S as a function of the number n of days that the person
works.
4. Evaluate the following functions at 𝑥 = −4.
a. 𝑓 𝑥 = 𝑥3
− 64
b. 𝑟 𝑥 = 5 − 𝑥
c. 𝑞 𝑥 =
𝑥+3
𝑥2+7𝑥+12
5. Let f and g be defined as 𝑓 𝑥 = 𝑥 − 5 and 𝑔 𝑥 = 𝑥2 − 1. Find
a. 𝑓 + 𝑔
b. 𝑓 − 𝑔
c. 𝑓. 𝑔
d. 𝑓/𝑔
