1. 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

2. f D E f is a function Graphical Illustration
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 f
Graphical Illustration
x *
z *
w *
5 *
* f(w)
* f(x)
* f(z)
* f(5) *
* 4
* - 9 D E
f is a function

3. More illustrations…. D E D E f is not a function Why?
x *
z *
w *
5 *
* f(w)
* f(x)
* f(z)
* f(5) *
* 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) *
* 4
* - 9 D E
f is not a function Why?
x in D has no values

4. 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.
Solution:
Note: f ( a + b ) f( a ) + f ( b )

5. 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

6. 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
f(x1) > f(x2)
f is constant
f(x1) = f(x2)
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

7. Example 1: Identify the interval(s) of the graph below where the function is
Increasing
Decreasing
Solution:
(a) Increasing
(b) Decreasing:

8. 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
decreasing -3

9. Interpretation of Domain and Range of a function f
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

10. 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 ,
Range =
Domain

11. More illustrations of Domain and Range of a graph of a function f
This graph does not end on both sides
Domain =
Range =
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 =
Range =

12. Class Exercise 1
Find the natural domain and range of the following graphs
Domain =
Range =
Domain =
Range =
Domain =
Domain =
Range =
Range =

13. Algebraic Approach to find the Domain of a function f
Example 1: Find the natural domain of the following functions
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
Domain =
(3) f is well defined when
Domain =
and
(4) f is well defined when
Domain = -5

14. Do all the Homework assigned in the syllabus for
Section 3.4