Functions Prepared by: Teresita P. Liwanag - Zapanta
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FunctionsPrepared by:
Teresita P. Liwanag - Zapanta
OBJECTIVES
•distinguish functions and relations
• identify domain and range of a function/relation
evaluate functions/relations.
•perform operation on functions/relations
•graph functions/relations
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.
DEFINITION
Ways of Expressing a Relation
5. Mapping2. Tabular form
3. Equation
4. Graph1. Set notation
.
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 x 0 1 2 3
y 0 2 4 6
3. Equation: y = 2x
4. Graph
y
x5-4 -2 1 3 5
5
-4
-2
1
3
5
-5 -1 4
-5
-1
4
-3
-5
2
2-5
-3
●
●
●1 2
2 4
63
0 0
x y
5. Mapping
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}
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.
),( ),0[
),( ),5[
),0[ ),(
2x
x2y
5.
Answer: D: all real nos. except -2
R: all real nos. except 2
1xy 6. Answer : D: all real nos. > –1 R: all real nos. > 0
g)
3x
x3y
7.Answer:D: all real nos. < 3R: all real nos. except 0
2except),(:D 2except),(:D
),1[:D ),0[:R
)3,(:D 0except),(:D
Exercises: Identify the domain and range of the following relations.
1. {(x,y) | y = x 2 – 4 }
8. y = (x 2 – 3) 2
x2
x3y)y,x(4.
3),( xyyx 2.
9),( xyyx3.
4x3xy)y,x( 2 5.
y = | x – 7 |6.
7. y = 25 – x 2
x
5x3y
9.
5x
25xy
2
10.
PROBLEM SET #5-1FUNCTIONS
Identify the domain and range of the following relations.
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.
xy sin13 xy
One-to-one and many-to-one functions
Each value of x maps to only
one value of y . . .
Consider the following graphs
Each value of x maps to only
one value of y . . .
BUT many other x values map
to that y.and each y is mapped from
only one x.
and
Functions
One-to-one and many-to-one functions
is an example of a one-to-one function
13 xy is an example of a many-to-one function
xy sin
xy sin13 xy
Consider the following graphs
and
Functions
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.
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
2x
x2y
e)
1xy f)
DEFINITION: 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 ”.
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 ).
4.If f(x) = x – 2 and g(x) = 2x2 – 3 x – 5 , Find: a) f(g(x)) b) g(f(x))
Piecewise Defined Function
if x<0
1x
x)x(f.1
2
0x if
2)2x(
1x
x3
)x(f.2
A piecewise defined function is defined by different formulas on different parts
of its domain. Example:
Piecewise Defined Function
if x<0 f(-2), f(-1), f(0), f(1), f(2)
EXAMPLE: Evaluate the piecewise function at the indicated values.
1x
x)x(f.1
2
0x if
2)2x(
1x
x3
)x(f.2
f(-5), f(0), f(1), f(5)
0x if
if
if
2x0 2x
DEFINITION: Operations on Functions
If f (x) and g (x) are two functions, thena)Sum and Difference
( f + g ) ( x ) = f(x) + g(x)
b)Product( f g ) ( x ) = [ f(x) ] [ g(x) ]
c)Quotient( f / g ) ( x ) = f(x) / g(x)
d) Composite ( f ◦ g ) ( x ) = f (g(x))
Example :1. Given f(x) = 11– x and g(x) = x 2 +2x –10
evaluate each of the following functions a. f(-5) b. g(2)c. (f g)(5)d. (f - g)(4)e. f(7)+g(x)f. g(-1) – f(-4)g. (f ○ g)(x)h. (g ○ f)(x)i. (g ○ f)(2)j. (f○ g)
)(x 2
•
•
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.
• We can also find the domain and range from the graph of a function.
Example: Graph each of the following functions.
5x3y.1
1xy.2
2x16y.3
5xy.4 2
3x2y.5
x
5x3y
6.
4xy.7
Graph of piecewise defined function
The graph of a piecewise function consists of separate functions.
1x2
x)x(f.1
2
if
if 1x
1x
3x
x9
x
)x(f.2 2
0x
1x 3x0
if
if
if
Example: Graph each piecewise function.
x
y
1-2
Plot the points in the coordinate plane
x
y
1-2
Plot the points in the coordinate plane
Graph of absolute value function.
Recall that
x
xx
if
if
0x 0x
Using the same method that we used in graphingpiecewise function, we note that the graph of f coincides with the line y=x to the right of the y axisand coincides with the line y= -x the left of the y-axis.
Example: Graph each of the follow functions.
y = | x – 7 |1.
y = x-| x - 2 |4.
x
y
1-2
Plot the points in the coordinate plane
•
Definition: Greatest integer function.
greatest integer less than or equal to x
The greatest integer function is defined by
x
Example: 0
1.0
3.0
9.0
1
1.1
2.1
9.1
2
1.2
4.3
4.3
9.0
0
0
0
0
1
1
1
1
2
2
3
-4
-1
Definition: Least integer function.
least integer greater than or equal to x
The least integer function is defined by
x
Example: 0
1.0
3.0
9.0
1
1.1
2.1
9.1
2
1.2
4.3
4.3
9.0
0
1
1
1
1
2
2
2
2
3
4
-3
0
Graph of greatest integer function.
xy Sketch the graph of
x xy 1x2
0x1 1x0 2x1 3x2
21
012
x
y
1-2
Plot the points in the coordinate plane
Graph of least integer function.
xy Sketch the graph of
x xy 1x2
0x1 1x0 2x1 3x2
10123
x
y
1-2
Plot the points in the coordinate plane
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