Functions

Prepared by Boipelo Radebe

Grade 10

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

Relations are set of ordered pairs

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.

Functions

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

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 ”.

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.

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.

Domain and range of a function

7 Function Families

What you need to know: Name

Equation

Domain

Range

Linear

Name – Constant

Equation –

Domain – (-,)

Range – [b]

y b

Linear

Name – Oblique Linear

Equation –

Domain – (-,)

Range – (-,)

y m x b

Power Functions

Name – Quadratic

Equation –

Domain – (-,)

Range – [0,)

y x 2

Reciprocal Functions

Name – Rational

Equation –

Domain –(-,0)(0,)

Range – (-,0) (0,)

yx

1

Power functions

Name - exponential

Equation – y= a

Domain – (-,)

Range – (0, )

x

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.

Graphs of functions?

Increasing and Decreasing Functions

A function f is increasing if:

A function f is decreasing if:

f x f x w hen

x x

( ) ( )1 2

1 2

f x f x w hen

x x

( ) ( )1 2

1 2

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

Transformations

Vertical ShiftHorizontal ShiftReflectingStretching/Shrinking

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

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

( )

( )

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

Use the rules of transformations to graph the following:

y x

y x

y x

y x

yx

2 3 2

1

24 3

2 6

1 3

1

25

2

3

( )

Trigonometric Functions

Name – Sine

Equation -y = a sin bx + c

Domain - (-,)

Range – [ 1. -1 ]

amplitude = a

period =b

360°

phase shift = bVertical shift

=c

Trigonometric Functions

Name – Cosine

Equation - y = a cos bx + c

amplitude = a

period =b

360°

phase shift = bVertical shift

=c

Domain - (-,)

Range – [ 1. -1 ]

Trigonometric Functions

Name – tangent (tan)

Equation -y = a tan bx + c

amplitude = a

period =b

180°

phase shift = bVertical shift

=c

Domain – x = - 180, -90, 90, 180

Range – (-,)

Graphs of functions in real life

Parabolas in life

Parabolic building

Do the following work on your own.

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))

Example 2Graph each of the following functions.

5x3y.1

1.2 xy

2x16y.3

5xy.4 2

3x2y.5

x

5x3y

4xy.7

6.

Example 3Determine 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

Reference Gurl, V . 2010. Afm chapter 4. functions.

http://www.slideshare.net/volleygurl22/afm-chapter-4-powerpoint?qid=e6cd91f5-5e87-4fa0-be23-f1afeb86873d&v=default&b=&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

Thank you