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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