This document discusses functions and their properties. It defines a function as a special relation where each first element is paired with exactly one second element. Functions are represented as sets of ordered pairs. The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values. Functions can be represented graphically and through equations, and can be transformed through shifts, reflections, and stretching/shrinking. Common function families include linear, quadratic, exponential, and trigonometric functions.

1. Functions
Prepared by Boipelo Radebe
Grade 10

2. Relation is referred to as any set of ordered pair.
Conventionally, It is represented by the ordered pair
( x , y ). x is called the first element or x-coordinate
while y is the second element or y-coordinate of the
ordered pair.
DEFINITIONDEFINITION

3. Relations are set of ordered
pairs

4. Definition: Function
•A function is a special relation such that every first
element is paired to a unique second element.
•It is a set of ordered pairs with no two pairs having
the same first element.

5. Functions
 Functions are relations, set of ordered pairs,in
which the first elements are not repeated.

6. Function Notation
•Letters like f , g , h and the likes are used to designate
functions.
•When we use f as a function, then for each x in the
domain of f , f ( x ) denotes the image of x under f .
•The notation f ( x ) is read as “ f of x ”.

7. Graph of a Function
•If f(x) is a function, then its graph is the set of all points
(x,y) in the two-dimensional plane for which (x,y) is an
ordered pair in f(x)
•One way to graph a function is by point plotting.
•We can also find the domain and range from the
graph of a function.

8. DEFINITION: Domain and RangeDEFINITION: Domain and Range
• All the possible values of x is called the domain.
• All the possible values of y is called the range.
• In a set of ordered pairs, the set of first elements
and second elements of ordered pairs is the
domain and range, respectively.

9. Domain and range of a
function

10. 7 Function Families
 What you need to know:
 Name
 Equation
 Domain
 Range

11. Linear
 Name – Constant
 Equation –
 Domain – (-∝,∝)
 Range – [b]
y b=

12. Linear
 Name – Oblique
Linear
 Equation –
 Domain – (-∝,∝)
 Range – (-∝,∝)
y m x b= +

13. Power Functions
 Name – Quadratic
 Equation –
 Domain – (-∝,∝)
 Range – [0,∝)
y x= 2

14. Reciprocal Functions
 Name – Rational
 Equation –
 Domain –(-∝,0)∪(0,∝)
 Range – (-∝,0) ∪ (0,∝)
y
x
=
1

15. Power functions
 Name - exponential
 Equation – y= a
 Domain – (-∝,∝)
 Range – (0, ∝)
x

16. Vertical Line Test
 A curve in the coordinate
plane is the graph of a
function if no vertical line
intersects the curve more
than once.

17. Graphs of functions?

18. Increasing and Decreasing
Functions A function f is increasing if:
 A function f is decreasing if:
f x f x w h e n
x x
( ) ( )1 2
1 2
<
<
f x f x w h e n
x x
( ) ( )1 2
1 2
>
<

19. State the intervals on which the function
whose graph is shown is increasing or
decreasing.

20. Transformations
Vertical Shift
Horizontal Shift
Reflecting
Stretching/Shrinking

21. General Rules for
Transformations
 Vertical shift:
 y=f(x) + c ⇒ c units up
 y=f(x) – c ⇒ c units down
 Horizontal shift:
 y=f(x+c) ⇒ c units left
 y=f(x-c) ⇒ c units right
 Reflection:
 y= – f(x) ⇒ reflect over x-axis
 y= f(-x) ⇒ reflect over y-axis
 Stretch/Shrink:
 y=af(x) ⇒ (a > 1) Stretch vertically
 y=af(x) ⇒ (0 < a < 1) Shrink vertically

22. Exploring transformations
 Graph
o Graph
o Graph
o Graph
y x= 2
y x
y x
y x
y x
y x
y x
= +
= −
= −
= +
=
=
2
2
2
2
2
2
3
2
4
3
2
1
2
( )
( )
{
{
{

23. Even & Odd Functions
 Algebraically:
 Even – f is even if f(-x) = f(x)
 Odd – f is odd if f(-x) = - f(x)
 Graphically:
 Even – f is even if its graph is symmetric to the y-
axis
 Odd – f is odd if its graph is symmetric to the
origin

24. Use the rules of transformations
to graph the following:
( )
y x
y x
y x
y x
y
x
= − +
= + −
= − −
= − + −
=
−
+
2 3 2
1
2
4 3
2 6
1 3
1
2
5
2
3
( )

25. Trigonometric Functions
Name – Sine
Equation -y = a sin bx + c
Domain - (-∝,∝)
Range – [ 1. -1 ]
amplitude = a
period =
b
360°
phase shift = b
Vertical shift =c

26. Trigonometric Functions
Name – Cosine
Equation - y = acosbx + c
amplitude = a
period =
b
360°
phase shift = b
Vertical shift =c
Domain - (-∝,∝)
Range – [ 1. -1 ]

27. Trigonometric Functions
Name – tangent (tan)
Equation -y = a tan bx + c
amplitude = a
period =
b
180°
phase shift = b
Vertical shift =c
Domain – x = - 180, -90, 90, 180
Range – (-∝,∝)

28. Graphs of functions in real life

29. Parabolas in life

30. Parabolic building

31. Do the following work on your
own.

32. EXAMPLE 1
Evaluate each function value
1. If f ( x ) = x + 9 , what is the value of f ( x 2
) ?
2. If g ( x ) = 2x – 12 , what is the value of g (– 2 )?
3. If h ( x ) = x 2
+ 5 , find h ( x + 1 ).
4.If f(x) = x – 2 and g(x) = 2x2
– 3 x – 5 ,
Find: a) f(g(x)) b) g(f(x))

33. Example 2
Graph each of the following
functions.
5x3y.1 −=
1.2 += xy
2
x16y.3 −=
5xy.4 2
−=
3
x2y.5 =
x
5x3
y
+
=
4xy.7 −−=
6.

34. Example 3
Determine Algebraically if the
function is even, odd or neither
y x x
y x x
y x x
y x x x
= +
= −
= −
= − + +
2
6 2
3
3 2
4
3 5
2 4 3 1

35. Reference
 Gurl, V . 2010. Afm chapter 4. functions.
http://www.slideshare.net/volleygurl22/afm-chapter-4-powerpoint?qid=e6c
from_search=1. Accessed 06 March 2014
 Manarang, K . 2011. 7 Functions.
http://www.slideshare.net/KathManarang/7-functions-
9175161. Accessed on 06 March 2014
 Farhana S .2013. Graphs and their functions.
http://www.slideshare.net/farhanashaheen1/function-
and-their-graphs-ppt?qid=e22cda30-fde3-4c4a-b233-
f00ff6f20596&v=default&b=&from_search=2. Accessed on
06 March 2014
 Schmitz, T .2008.Higher Maths 1.2.3 - Trigonometric
Functions. http://www.slideshare.net/timschmitz/higher-
maths-123-trigonometric-functions-358346?qid=4e5bcb29-
5942-48aa-9735-bf4c30ac5f05&v=qf1&b=&from_search=1.
Accessed on 06 March 2014

36. Thank you