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• 1. Functions
Prepared by:
Teresita P. Liwanag - Zapanta
• 2. OBJECTIVES
• distinguish functions and relations
• 3. identify domain and range of a function/relation
evaluate functions/relations.
• perform operation on functions/relations
• 4. graph functions/relations
• DEFINITION
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.
• 5. Ways of Expressing a Relation
4. Graph
1. Set notation
5. Mapping
2. Tabular form
3. Equation
• 6. Example: Express the relation y = 2x;x= 0,1,2,3
in 5 ways.
.
1. Set notation
(a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) } or
(b) S = { (x , y) such that y = 2x, x = 0, 1, 2, 3 }
2. Tabular form
• 7. y
5
5
4
3
2
1
x
5
-4
-2
1
3
5
-5
-1
4
-3
2
-5
-1
-2
-3
-4
-5
-5
3. Equation: y = 2x
5. Mapping
4. Graph
x
y

0
0

1
2

2
4
6
3
• 8. DEFINITION: Domain and Range
All the possible values of x is called the domain and 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.
Example: Identify the domain and range of the following
relations.
1.) S = { ( 4, 7 ),( 5, 8 ),( 6, 9 ),( 7, 10 ),( 8, 11 ) }
Answer : D: { 4,5,6,7,8} R:{7,8,9,10,11}
• 9. 2.) S = { ( x , y ) s. t. y = | x | ; x  R }
Answer: D: all real nos.R: all real nos. > 0
3) y = x 2 – 5
Answer. D: all real nos. R: all real nos. > -5
4) | y | = x
Answer: D: all real nos. > 0 R: all real nos.
• 10. g)
D: all real nos. except -2
R: all real nos. except 2
5.
D: all real nos. > –1
R: all real nos. > 0
6.
D: all real nos. < 3
R: all real nos. except 0
7.
• 11. Exercises: Identify the domain and range of the
following relations.
1. {(x,y) | y = x 2 – 4 }
2.
5.
7. y = 25 – x 2
y = | x – 7 |
6.
4.
3.
8. y = (x 2 – 3) 2
9.
10.
• 12. PROBLEM SET #5-1
FUNCTIONS
Identify the domain and range of the following relations.
• 13. Definition: Function
• A function is a special relation such that every first element is paired to a unique second element.
• 14. It is a set of ordered pairs with no two pairs having the same first element.
• Functions
One-to-one and many-to-one functions
Consider the following graphs
and
Each value of x maps to only one value of y . . .
Each value of x maps to only one value of y . . .
and each y is mapped from only one x.
BUT manyother x values map to that y.
• 15. is an example of a many-to-one function
is an example of a one-to-one function
Functions
One-to-one and many-to-one functions
Consider the following graphs
and
One-to-many is NOT a function. It is just a relation. Thus a function is a relation but a relation could never be a function.
• 16. Example: Identify which of the following relations are functions.
a) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) }
b) S = { ( x , y ) s. t. y = | x | ; x  R }
c) y = x 2 – 5
d) | y | = x
e)
f)
• 17. DEFINITION: Function Notation
• Letters like f , g , h and the likes are used to designate functions.
• 18. 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 .
• 19. The notation f ( x ) is read as “ f of x”.
• EXAMPLE: 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 ).
If f(x) = x – 2 and g(x) = 2x2 – 3 x – 5 ,
Find: a) f(g(x)) b) g(f(x))
• 20. Piecewise Defined Function
A piecewise defined function is defined by different formulas on different parts of its domain.
Example:
if x<0
if
• 21. Piecewise Defined Function
EXAMPLE: Evaluate the piecewise function at the
indicated values.
if x<0
f(-2), f(-1), f(0), f(1), f(2)
if
if
if
if
f(-5), f(0), f(1), f(5)
• 22. DEFINITION: Operations on Functions
If f (x) and g (x) are two functions, then
Sum and Difference
( f + g ) ( x ) = f(x) + g(x)
Product
( f g ) ( x ) = [ f(x) ] [ g(x) ]
Quotient
( f / g ) ( x ) = f(x) / g(x)
d) Composite
( f ◦ g ) ( x ) = f (g(x))
• 23. Example :1. Given f(x) = 11– x and g(x) = x 2 +2x –10
evaluate each ofthe following functions
f(-5)
g(2)
(f g)(5)
(f - g)(4)
f(7)+g(x)
g(-1) – f(-4)
(f ○ g)(x)
(g ○ f)(x)
(g ○ f)(2)
(f○ g)
• 24.
• 25.
• 26. DEFINITION: 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.
• 27. We can also find the domain and range from the
graph of a function.
• 28. Example: Graph each of the following functions.
6.
• 29. Graph of piecewise defined function
The graph of a piecewise function consists of separate functions.
Example: Graph each piecewise function.
if
if
if
if
if
• 30. Plot the points in the coordinate plane
y
x
-2
1
• 31. Plot the points in the coordinate plane
y
x
-2
1
• 32. Graph of absolute value function.
Recall that
if
if
Using the same method that we used in graphing
piecewise function, we note that the graph of f
coincides with the line y=x to the right of the y axis
and coincides with the line y= -x the left of the y-axis.
• 33. Example: Graph each of the follow functions.
y = | x – 7 |
y = x-| x - 2 |
1.
4.
• 34. Plot the points in the coordinate plane
y
x
-2
1
• 35.
• 36. Definition: Greatest integer function.
The greatest integer function is defined by
greatest integer less than or equal to x
Example:
1
3
0
-4
1
0
-1
1
0
2
0
2
1
• 37. Definition: Least integer function.
The least integer function is defined by
least integer greater than or equal to x
Example:
2
4
0
-3
2
1
0
2
1
2
1
3
1
• 38. Graph of greatest integer function.
Sketch the graph of
• 39. Plot the points in the coordinate plane
y
x
-2
1
• 40. Graph of least integer function.
Sketch the graph of
• 41. y
Plot the points in the coordinate plane
x
1
-2