teaching the basic of functions in mathematics.

1. Functions
Students will determine if a given equation is a function using the
vertical line test and evaluate functions given member(s) of the domain.
Compiled by : Motlalepula Mokhele
Student at the University of Johannesburg (2014)

3. 1 
5
2

6
3

7
4

8
5 
9
The Rule is ‘ADD 4’

4. Ahmed 
Paris
Peter 
London
Ali
Dubai
Jaweria

New York
Cyprus
Hamad 
Has Visited
There are MANY arrows from each person and each place is related to MANY
People. It is a MANY to MANY relation.

5. Person
Bilal
Has A Mass of

Kg
62
Peter

Salma 
Alaa

George 
Aziz
64
66

In this case each person has only one mass, yet several people have the same
Mass. This is a MANY to ONE relationship

6. Is the length of
cm

14 
object
Pen
Pencil
Ruler

30 
Needle
Stick
Here one amount is the length of many objects.
This is a ONE to MANY relationship

7. FUNCTIONS

Many to One Relationship

One to One Relationship

8. x2x+1
A
B
0
1
2
3
4
Domain
1
2
3
4
5
6
7
8
9
Co-domain
Image Set (Range)

9. f : x  x2 +
4
f(
x)
=
x2 +
4
The upper function is read as follows:‘Function f such that x is mapped onto x2+4

10. Lets look at some function
Type questions
If
f ( x ) = x 2 + 4 and g ( x ) = 1 − x 2
F ind f ( 2 )
F ind g ( 3 )
2
f(x) = x 2 + 4
2
=8
g(x) = 1 − x 2
3
3
= -8

11. Consider the function f(x) = 3x − 1
x
We can consider this as two simpler
functions illustrated as a flow diagram
3x
Multiply by 3
Subtract 1
3x − 1
Consider the function f : x  (2x + 5) 2
x
Multiply by 2
2x
Add 5
2x + 5
Square
(2x + 5) 2

12. Consider 2 functions
f : x  3x + 2 and g(x) : x  x 2
fg is a composite function, where g is performed first and then f is performed
on the result of g.
The function fg may be found using a flow diagram
x
square
g
Thus fg = 3x 2 + 2
x2
Multiply by 3
3x 2
f
Add 2
3x 2 + 2

13. x2
3x + 2
f
g
2
4
2
fg (x)
3x 2 + 2
14

14. Consider the function
f(x) = 5x − 2
3
Here is its flow diagram
5 x -2
5x
x
Multiply by 5
Subtract 2
f(x) = 5x − 2
3
Divide by three
Draw a new flow diagram in reverse!. Start from the right and go left…
3 x +2
5
3x
3 x +2
Divide by 5
And so
f −1 (x) = 3x + 2
5
Add two
x
Multiply by three

15. (b)
(a)
(c)
(d)
(a) and (c)

16. (b)
(a)
(c)
(d)
(a) and (c)

17. Functions and Their
Graphs

18. Definition of Relation
 Relation
– a set of ordered pairs, which contains
the pairs of abscissa and ordinate. The first
number in each ordered pair is the x-value or
the abscissa, and the second number in each
ordered pair is the y-value, or the ordinate.
 Domain
is the set of all the abscissas, and range
is the set of all ordinates.

19. Relations

A relation may also be shown using a table of values or
through the use of a mapping diagram.

Illustration:

Using a table.
Domain
0
1
2
3
4
7
Using a mapping diagram.
Range
1
2
3
4
5
8

20. Definition of Function
 Function
– a characteristic of set of
values where each element of the
domain has only one that
corresponds with it in the range. It
is denoted by any letter of the
English alphabet.
 The function notation f(x) means
the value of function f using the
independent number x.

21. Example 1a.
 Given
the ordered pairs below, determine if it
is a mere relation or a function.

(0,1) , (1, 2), (2, 3), (3, 4), (4, 5), (7, 8)
 Answer:
 For
every given x-value there is a
corresponding unique y-value. Therefore, the
relation is a function.

22. Example 1b.
Which relation represents a function?
A. {(1,3), (2, 4), (3,5), (5, 1)}
B. {(1, 0), (0,1), (1, -1)}
C. {(2, 3), (3, 2), (4, 5), (3, 7)}
D. {(0, 0), (0, 2)}

Answer:
A


23. Example 1c.

Which mapping diagram does not represent a function?

A.
B.

C.
D.

24. Types of Functions

25. 1.Increasing, Decreasing, and Constant
Functions
A function is increasing on an interval if for any x1, and x2 in the
interval, where x1 < x2, then f (x1) < f (x2).
A function is decreasing on an interval if for any x1, and x2 in the
interval, where x1 < x2, then f (x1) > f (x2).
A function is constant on an interval if for any x1, and x2 in the
interval, where x1 < x2, then f (x1) = f (x2).
(x2, f (x2))
(x1, f (x1))
Increasing
f (x1) < f (x2)
(x1, f (x1))
(x2, f (x2))
(x1, f (x1))
(x2, f (x2))
Decreasing
f (x1) < f (x2)
Constant
f (x1) < f (x2)

26. Example 8a.
Describe the increasing, decreasing, or constant behavior of each
function whose graph is shown.
a
b
5
.
4
5
.
3
4
3
2
1
1
-5 -4 -3 -2 -1
-1
-2
1
2
3 4
5
-5 -4 -3 -2 -1
-1
-2
-3
2
3 4
5
-3
-4
-5
1
-4
-5
Solutio
a. n
The function is decreasing on the interval (-∞, 0), increasing on the
interval (0, 2), and decreasing on the interval (2, ∞).

27. Example 8a.
Describe the increasing, decreasing, or constant behavior of each function
whose graph is shown.
a
b
5
.
4
5
.
3
4
3
2
1
1
-5 -4 -3 -2 -1
-1
-2
1
2
3 4
5
-5 -4 -3 -2 -1
-1
-2
-3
2
3 4
5
-3
-4
-5
1
-4
-5
Solution:
b.
• Although the function's equations are not given, the graph
indicates that the function is defined in two pieces.
• The part of the graph to the left of the y-axis shows that the
function is constant on the interval (-∞, 0).
• The part to the right of the y-axis shows that the function is
increasing on the interval [0,∞).

28. Example 8b.
Describe the
increasing,
decreasing, or
constant behavior of
each function whose
graph is shown.
Decreasing on (-∞, 0);
Increasing on (0, ∞)

29. Example 8b.
Describe the
increasing,
decreasing, or
constant behavior of
each function whose
graph is shown.
Increasing on (-∞, 2);
Constant on (2, ∞)

30. Example 8c.
Describe the
increasing,
decreasing, or
constant behavior of
each function whose
graph is shown.
Increasing on (-∞,∞)

31. 2.Continuous and Discontinuous Functions
A continuous function is represented by a graph
which may be drawn using a continuous line or curve,
while a discontinuous function is represented by a
graph which has some gaps, holes or breaks
(discontinuities).

32. 3.Periodic Functions
A periodic function is a function whose values repeat
in periods or regular intervals.
y = tan(x)
y = cos(x)

33. A linear function is a function of the form f(x) = mx +b
where m and b are real numbers and m ≠ 0.
Domain: the set of real numbers
Range: the set of real numbers
Graph: straight line
Example: f(x) = 2 - x

34. 5. Quadratic Functions
A quadratic function is a function of the form f(x) = ax2
+bx +c where a, b and c are real numbers and a ≠ 0.
Domain: the set of real numbers
Graph: parabola
Examples: parabolas
parabolas
opening upward
opening downward

35. Graphs of Quadratic
Functions
The graph of any quadratic function is called a parabola.
Parabolas are shaped like cups, as shown in the graph
below.
If the coefficient of x2 is positive, the parabola opens
upward; otherwise, the parabola opens downward.
The vertex (or turning point) is the minimum or maximum
point.

36. Evaluation of Functions

37. Example 2.
If f (x) = x2 + 3x + 5, evaluate:
a. f (2)
b. f (x + 3)
c. f (-x)
Solution
a. We find f (2) by substituting 2 for x in the
equation.
f (2) = 22 + 3 • 2 + 5 = 4 + 6 + 5 = 15
Thus, f (2) = 15.

38. Example 2.
If f (x) = x2 + 3x + 5, evaluate: b. f (x + 3)
Solution
b. We find f (x + 3) by substituting (x + 3) for x in
the equation.
f (x + 3) = (x + 3)2 + 3(x + 3) + 5
Equivalently,
f (x + 3) = (x + 3)2 + 3(x + 3) + 5
= x2 + 6x + 9 + 3x + 9 + 5
= x2 + 9x + 23.

39. Example 2.
If f (x) = x2 + 3x + 5, evaluate: c. f (-x)
Solution
c. We find f (-x) by substituting (-x) for x in the
equation.
f (-x) = (-x)2 + 3(-x) + 5
Equivalently,
f (-x) = (-x)2 + 3(-x) + 5
= x2 –3x + 5.

40. Example 3a.
 Which
is the range of the relation described
by y = 3x – 8 if its domain is {-1, 0, 1}?
 A)
{-11, 8, 5}
 B)
{-5, 0 5}
 C)
{-11, -8, -5}
 D)
{0, 3, 5}

41. Operations on Functions

42. Sum, Difference, Product, and Quotient of
Functions
Let f and g be two functions. The sum, the
difference, the product , and the quotient are
functions whose domains are the set of all real
numbers common to the domains of f and g,
defined as follows:
 Sum:
(f + g)(x) = f (x)+g(x)
 Difference:
(f – g)(x) = f (x) – g(x)
 Product:
(f • g)(x) = f (x) • g(x)
 Quotient:
(f / g)(x) = f (x)/g(x), g(x) ≠ 0

43. Example 4a.
Let f(x) = 2x+1 and g(x) = x2 - 2.
Find
a. (f + g) (x) c.(g – f) (x)
e. (f / g) (x)
b. (f – g) (x)
d. (f ∙ g) (x)
f. (g/f) (x)
Solution:
a. (f + g) (x) = f(x) + g( x) = (2x+1 )+ (x2 – 2) = x2 + 2x - 1
b. (f – g)(x) = f(x) - g(x) = (2x+1) - (x 2 - 2) = -x2 + 2x + 3
c. (g – f)(x) = g(x) - f(x) = (x2 - 2) – (2x +1) = x2 - 2x - 3
d. (f ∙ g)(x) = f(x) ∙ g(x) = (2x+1)(x 2 - 2) = 2x3 + x2 - 4x - 2
e. (f/g)(x) = f(x)/g(x) = (2x+1)/(x 2 - 2),
x≠± 2
2
f. (g/f)(x) = g(x)/f(x) = (x - 2)/(2x +1), 1
x≠−
2

44. Example 5a.
Given f (x) = 3x – 4 and g(x) = x2 + 6,
find:
a. (f ○ g)(x) b. (g ○ f)(x)
Solution
a. We begin with (f o g)(x), the composition of f with g. Because (f o
g)(x) means f (g(x)), we must replace each occurrence of x in the
equation for f by g(x).
f (x) = 3x – 4
(f ○ g)(x) = f (g(x)) = 3(g(x)) – 4
= 3(x2 + 6) – 4
= 3x2 + 18 – 4
= 3x2 + 14
Thus, (f ○ g)(x) = 3x2 + 14.

45. Example 5a.
Given f (x) = 3x – 4 and g(x) = x2 + 6,
find:
a. (f ○ g)(x) b. (g ○ f)(x)
Solution
b. Next, we find (g o f )(x), the composition of g with f.
Because (g o f )(x) means g(f (x)), we must replace each
occurrence of x in the equation for g by f (x).
g(x) = x2 + 6
(g ○ f )(x) = g(f (x)) = (f (x))2 + 6
= (3x – 4)2 + 6
= 9x2 – 24x + 16 + 6
= 9x2 – 24x + 22
Notice that (f ○ g)(x) is not the same as (g ○ f )(x).

46. Graphs of Relations and Functions

47. Graph of a Function

If f is a function, then the graph of f is the set of all
points (x,y) in the Cartesian plane for which (x,y) is an
ordered pair in f.

The graph of a function can be intersected by a vertical
line in at most one point.

Vertical Line Test

If a vertical line intersects a graph more than once, then
the graph is not the graph of a function.

48. Example 6a.

Determine if the graph is a graph of a function or just a graph of a relation.
8
6
4
2
5
-2
-4
10
15
graph
of a
relation

49. Example 6b.

Determine if the
graph is a graph of a
function or just a
graph of a relation.
graph
of a
function

50. Example 6c.

Determine if the graph is
a graph of a function or
just a graph of a relation.
graph
of a
relation

51. Example 6d.

Determine if the
graph is a graph of a
function or just a
graph of a relation.
16
14
12
10
8
6
4
graph
of a
relation
2
A
15
10
5
5
2
4
6
8
10
15
20
25

52. Example 6e.

Determine if the graph is a graph of a function or just a graph of a
relation.
4
3
2
1
-6
-4
-2
2
-1
-2
-3
-4
4
6
graph
of a
relation

53. Example 6f.

Determine if the graph is a graph of a function or just a
graph of a relation.
6
4
2
-10
-5
5
-2
-4
-6
10
graph
of a
relation

54. Example 6g.

Determine if the graph is a graph of a function or just a
graph of a relation.
3
1
-3 -2 -1
-1
-2
-3
-5
1 2
3 4
graph
of a
function

55. Graphing Parabolas

Given
4.
Find any x-intercepts by replacing f (x) with 0. Solve
the resulting quadratic equation for x.
5.
Find the y-intercept by replacing x with zero.
6.
Plot the intercepts and vertex. Connect these points
with a smooth curve that is shaped like a cup.
f(x) = ax2 + bx +c

56. Graphing Parabolas

Given
1.
Determine whether the parabola opens upward or downward.
If a > 0, it opens upward. If a < 0, it opens downward.
2.
Determine the vertex of the parabola. The vertex is
f(x) = ax2 + bx +c
 − b 4ac − b 2 
,


4a 
 2a

The axis of symmetry is
−b
x=
2a
The axis of symmetry divides the parabola into two equal parts such that one part
is a mirror image of the other.

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