Mathematics in Modern World

1. Relation and Function

2. Objectives:
 Define relation and function;
 Identify different kinds of relation;
 Determine the function in a relation; and
 Evaluate and perform operations on
functions.

3. Relation and Function
Relations and Functions” are the most important topics in
algebra and these are the two different words having
different meanings mathematically.
 In an ordered pair is represented as (INPUT, OUTPUT):
The relation shows the relationship between INPUT and
OUTPUT. Whereas, a function is a relation which derives one
OUTPUT for each given INPUT.
(x,y)

4. In an ordered pair we have:
Domain
It is a collection of the first values in the ordered pair (Set of all
input (x) values).
Range
It is a collection of the second values in the ordered pair (Set of all
output (y) values).
Example:
In the relation, {(-2, 3), (4, 5), (6, -5), (-2, 5)},
The elements in the domain are {-2, 4, 6}
The elements in the range are {-5, 3, 5}.
Note: Don’t consider duplicates while writing the domain and range

5.  In mathematics relation is a subset of the
Cartesian product. Or simply, a bunch of points
(ordered pairs). In other words, the relation
between the two sets is defined as the collection
of ordered pairs, in which the ordered pair is
formed by the object from each set.
 Example: {(-2, 1), (4, 3), (7, -3)}, usually written
in set notation form with curly brackets.

6. Relation Representation

7. Kinds of Relation
 One to One
 One to Many
 Many to One
1
0
7
2
3
5
1
-3
2
-5
3
5
-6
-7
5
-3
1

8. What is a Function?
 A function is a relation which describes that there should
be only one output for each input (or) we can say that a
special kind of relation (a set of ordered pairs), which
follows a rule i.e. every x-value should be associated
with only one y-value is called a function.
 For example:
Domain Range
-1 -3
1 3
3 9
2 9
function

9. Which of the relation is a function?
 One to One
 One to Many
Domain Range
2 1
3 0
5 7
Domain Range
3 1
3 -3
5 2
5 -5
function
not a
function

10.  Many to One
Domain Range
3 3
2 3
-1 3
-2 4
function

11. Note: All functions are relations, but not all
relations are functions.

12. How to determine a graph whether a function
or not?
 To determine a graph whether a function or not, we use
the “vertical line test”.
 The vertical line test is a method that is used to
determine whether a given relation is a function or not.
The approach is rather simple. Draw a vertical line cutting
through the graph of the relation, and then observe the
points of intersection.
 If a vertical line intersects the graph in all places at
exactly one point, then the relation is a function.

13. Determine the following is a function or not.
1. Graph of y = x + 1 2. Graph of y = x2
– 2.
function function

14. 3. Graph of y = x3
4. Graph of x = y2
function
not a function

15. 5. Graph of x2
+ y2
= 9. 6. Graph of x = y3
– y + 2
not a function not a function

16. Naming a Function
 To name a function, we use letters in the alphabet in lower case.
The most commonly used letters are f, g, h to name a function.
1.) y = x2
– x - 2 2.) y = 3c + 8 3.) y = m2
+ 6m + 5
If you choose f to name #1, then you have f(x) = y. f(x)
can be read as f of x. why x? it is because that’s the
variable you’ve seen in the right side of the equation.
1.) f(x) = x2
– x - 2 2.) g(c) = 3c + 8 3.) h(m) = m2
+ 6m + 5

17. Evaluation of Function
 To evaluate a function is just simply substitute the value
of the variable and then simplify.
 Given the following functions:
f(x) = 3x - 8
g(x) = 4x + 1
h(x) = x² - 2x + 5
j(x) = (6x + 5)/7
find:
1.) f(5)
What is f(5)?
Because the name of a function
is f so we take;
f(x) = 3x - 8
Substitute 5 as value of x in
f(x).
f(5) = 3(5) – 8
Then simplify.
=15 – 8
f(5) = 7

18. f(x) = 3x - 8
g(x) = 4x + 1
h(x) = x² - 2x + 5
j(x) = (6x + 5)/7
2.) h(3)
h(x) = x² - 2x + 5
h(3) = (3)² - 2(3) + 5
= 9 – 6 + 5
= 3 + 5
h(3) = 8
3.) g(0.5)
g(x) = 4x + 1
g(0.5) = 4(0.5) + 1
= 2 + 1
g(0.5) = 3

19. f(x) = 3x - 8
g(x) = 4x + 1
h(x) = x² - 2x + 5
j(x) = (6x + 5)/7
4.) j(5)
j(x) = (6x + 5)/7
j(5) = (6(5) + 5)/7
= (30 + 5)/7
= 35/7
j(5) = 5
5.) f(2x – 1)
f(x) = 3x – 8
f(2x – 1) = 3(2x -1) – 8
= 6x - 3 – 8
f(2x – 1) = 6x -11

20. Operations on Functions
Here are the following operations on functions:
1.) Addition
(f + g) (x) = f(x) + g(x)
example:
f(x) = 3x – 4
g(x) = 5x + 1
What is (f + g) (x)?
(f + g) (x) = f(x) + g(x)
Substitute the given function to the formula and
be sure you put a parenthesis on it.
(f +g) (x) = (3x – 4) + (5x + 1)
Then simplify. (multiply + sign to (5x +1))
= 3x – 4 + 5x + 1
= 3x + 5x – 4 + 1
(f + g) (x) = 8x - 3

21. 2.) Subtraction
(f – g) (x) = f(x) – g(x)
example:
f(x) = 3x – 4
g(x) = 5x + 1
What is (f – g) (x)?
(f – g) (x) = f(x) – g(x)
Substitute the given function to the
formula and be sure you put a parenthesis
on it.
(f – g) (x) = (3x – 4) – (5x + 1)
Then simplify. (multiply – sign to (5x + 1))
=3x – 4 - 5x – 1
= 3x – 5x - 4 – 1
(f – g) (x) = -2x - 5

22. 3.) Multiplication
(fg) (x) = f(x)  g(x)
example:
f(x) = 3x – 4
g(x) = 5x + 1
What is (fg) (x)?
(fg) (x) = f(x)  g(x)
Substitute the given function to the
formula and be sure you put a parenthesis
on it.
(fg) (x) = (3x – 4)  (5x + 1)
Then simplify. (you can use the foil
method)
= 15x² + 3x – 20x – 4
(fg) (x) = 15x² - 17 x - 4

23. 4.) Division
(f/g) (x) = f(x) / g(x)
example:
f(x) = x² + 2x + 1
g(x) = x + 1
9/3
3(3)/3
3/1=3
What is (f/g) (x)?
(fg) (x) = f(x) / g(x)
Substitute the given function to the formula
and be sure you put a parenthesis on it.
(f/g) (x) = (x² + 2x + 1) / (x + 1)
Then simplify. (you can use factoring method
–express the numerator by its factors)
= {(x + 1) (x + 1)} / (x + 1)
Cancel one group of (x + 1) in both
numerator and denominator.
(f/g) (x) = x + 1

24. 5.) Composite of a Function
(f o g) (x) = f (g(x))
example:
f(x) = 3x – 4
g(x) = 5x + 1
What is (f o g) (x)?
(f o g) (x) = f(g(x))
The value of x in f(x) is the g(x)
f(x) = 3x – 4
f(g(x)) = 3(g(x)) - 4
Substitute the function g(x) to the value
of x in f(x) and be sure you put a
parenthesis on it.
f(5x + 1) = 3(5x + 1) - 4
Then simplify.
= 15x + 3 – 4
[f(g)] (x) = 15x -1