دروس في الرياضيات العامة الجامعية .. النطاق والمدى
TRANSCRIPT
Section 3.4
Objectives:
• Find function values
• Use the vertical line test
• Define increasing, decreasing and constant functions
•Interpret Domain and Range of a function Graphically and Algebraically
Function: A function f is a correspondence from a set D to a set E that assigns to each element x of D exactly one value ( element ) y of E
Graphical Illustration
E
x *
z *
w *
5 *
* f(w)
* f(x)
* f(z)
* f(5)
* 3* 4* - 9
D
f
f is a function
More illustrations….
x *
z *
w *
5 *
* f(w)
* f(x)
* f(z)
* f(5)
* 3* 4* - 9
D
E
f is not a function Why?
x in D has two values
x *
z *
w *
5 *
* f(w)
* f(x)
* f(z)
* f(5)
* 3* 4* - 9
D
E
f is not a function Why?
x in D has no values
Find function values
Example 1: Let f be the function with domain R such that f( x) = x2 for every x in R.
( i ) Find f ( -6 ), f ( ), f( a + b ), and f(a) + f(b) where a and b are real numbers.3
Solution: 3666 2 f
3332f
222 2 babababaf
22 babfaf Note: f ( a + b ) f( a ) + f ( b )
Vertical Line Testof functions
Vertical Line test: The graph of a set of points in a coordinate plane is the graph of a function if every vertical line intersects the graph in at most one point
Example: check if the following graphs represent a function or not
Function
Function
FunctionNot Function
Increasing, Decreasing and Constant Function
Terminology Definition Graphical Interpretation
f is increasing
over interval I
f(x1) < f(x2)
whenever
x1 < x2
f is decreasing
over interval I
f(x1) > f(x2)
whenever
x1 < x2
f is constant
over interval I
f(x1) = f(x2)
whenever
x1 = x2
x1 x2
f(x1)f(x2)
x
y
x1 x2
f(x1) f(x2)
x
y
x1 x2
f(x1) f(x2)
x
y
Example 1: Identify the interval(s) of the graph below where the function is
(a) Increasing
(b) Decreasing
Solution:
(a) Increasing ,20,
(b) Decreasing: 2,0
Example 2: Sketch the graph that is decreasing on ( ,- 3] and [ 0, ), increasing on [ -3 ,0 ],
f(-3) = 2 and f (2 ) = 0
Solution:
-3 0
decreasing increasing decreasing
Interpretation of Domain and Rangeof a function f
Domainis the
Set of all x where f is
well defined
Rangeis the set of all values
f( x )Where x is in the
domain
f
Graphical Approach toDomain and Range
Example 1: Find the natural domain and Range of the graph of the function f below
The function f represents f (x ) = x2. f is well defined everywhere in R. Therefore,
Domain = R ),(
Range
Domain
Every value of f is non-negative ( greater than or equal to 0. Therefore ,
Range = ),[ o
More illustrations of Domain and Range of a graph of a function f
These two graphs seem similar, but the domain and range are different
This graph does not end on both sides
Domain = ),(
Range = ),0[
This graph ends, it is also not defined at x = –2 and
well defined at x =2
Domain =
Range =
]2,2(
]4,0[
Class Exercise 1 Find the natural domain and range of the following graphs
Domain = Range = Domain =
Domain = Domain =
Range =
Range =
Range =
)2,2[ ]2,0[ ,R ]1,1[
,33,33,
,R
]25.5,75.0()25.2,75.6[
]3,75.0(3
Algebraic Approach to find theDomain of a function f
Example 1: Find the natural domain of the following functions
13
1024
102
133
1022
)(131
x
xxg
x
xxf
functionrootSquarexxf
functionLinearxxf
Solution:
( 1 ) f is a linear function. f is well-defined for all x. Therefore, Domain = R
( 2 ) f is a square root function. f is well defined when
0 10 2 x
5x Domain = ),5[
(3) f is well defined when
0 10 2 x 5x Domain = ),5(
(4) f is well defined when
0 10 2 x and 013 x
-5 3/1
Domain = ),3/1()3/1,5[
Do all the Homework assigned in the syllabus for
Section 3.4