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