Transcript Document

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
f
* f(w)
x *
* f(x)
z *
*
w *
* f(5)
5 *
* 3
* 4
* -9
f(z)
D
E
f is a function
More illustrations….
* f(w)
f is not a function Why?
x *
* f(x)
z *
*
w *
* f(5)
5 *
* 3
* 4
* -9
D
f(z)
x in D has two values
E
* f(w)
f is not a function Why?
x *
* f(x)
z *
*
w *
* f(5)
5 *
* 3
* 4
* -9
D
E
f(z)
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
Solution:
f ( -6 ),
f(
3),
f( a + b ), and f(a) + f(b) where a and b are real numbers.
f  6   6  36
2
f
 3  3
2
3
f a  b  a  b
2
f a   f b  a 2  b2
 a 2  2ab  b2
Note: f ( a + b )

f( a ) + f ( b )
Vertical Line Test
of 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
Not Function
Function
Increasing, Decreasing and Constant Function
Terminology
Definition
f is increasing
over interval I
f(x1) < f(x2)
whenever
x1 < x2
Graphical Interpretation
y
f(x1)

x1
f is decreasing
over interval I
f(x1) > f(x2)
whenever
x1 < x2

x1
f(x1) = f(x2)
whenever
x1 = x2
f(x2)
x2
x
y
f(x1)
f is constant
over interval I

f(x )
2
x2
x
y
f(x1)
 
x1
x2
f(x2)
x
Example 1: Identify the interval(s) of the graph below where the function is
(a)
Increasing
(b)
Decreasing
Solution:
(a) Increasing
 ,0  2, 
(b) Decreasing:
0,2 
Example 2: Sketch the graph that is decreasing on (  ,- 3] and [ 0,  ), increasing on [ -3 ,0 ],
f(-3) = 2 and f (2 ) = 0
Solution:
decreasing
increasing
-3
decreasing
0
Interpretation of Domain and Range
of a function f
f
Domain
is the
Set of all x
where f is
well defined
Range
is the set of all values
f( x )
Where x is in the
domain
Graphical Approach to
Domain and Range
Example 1: Find the natural domain and Range of the graph of the function f below
Range
The function f represents f (x ) = x2. f is well defined
everywhere in R. Therefore,
Domain = R
 (,)
Every value of f is non-negative ( greater than or equal
to 0. Therefore ,
[o,)
Range =
Domain
More illustrations of Domain and Range of a graph of a function f
This graph does not end on both sides
Domain = (,)
Range = [0,)
These two graphs seem similar, but the domain
and range are different
This graph ends, it is also
not defined at x = –2 and
well defined at x =2
Domain = (2,2]
Range = [0,4]
Class Exercise 1
Domain =
Domain =
Range =
[2,2)
Find the natural domain and range of the following graphs
Range =
 ,3   3,3  3, 
R   , 
[0,2]
Domain = R 
 , 
Range = [1,1]
Domain = [6.75,2.25)  (0.75,5.25]
Range =  3 (0.75,3]
Algebraic Approach to find the
Domain of a function f
Example 1: Find the natural domain of the following functions
1 f  x   3 x  1 ( Lin ea r fu n ctio n )
2  f  x   2 x  1 0 S q u a re ro o t fu n ctio n
3
4 
f x  
g x  
3x  1
2x 10
2x 10
3x  1
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
2 x  10  0
(3) f is well defined when
2 x  10  0
(4) f is well defined when
-5
x  5
Domain = [ 5, )
x  5
Domain = ( 5, )
2 x  10  0 and
3x  1  0
Domain = [5,1 / 3)  (1 / 3, )
1/ 3
Do all the Homework assigned in the syllabus for
Section 3.4