Lesson 2 - Functions and their Graphs - NOTES.ppt

Functions and their Graphs
Lesson 2
Sections 1.2 and 1.3

Directions for Open–Minded
Questions – Warm Up
 Everyone silently read through both problems.
 Write down what you know.
 Read the two questions again.
 Group Discuss
 You have 5 minutes to read and discuss and
10 minutes or less to show solutions.

Solve the following. Try to find
multiple solution paths!
 In an all-adult apartment building, 2/3 of the men are
married to 3/5 of the women. What fraction of the
residents is married?
 A farmer had hens and rabbits. These animals have
50 heads and 140 feet. How many hens and rabbits
does the farmer have?

In an all-adult apartment building, 2/3 of the men are married to
3/5 of the women. What fraction of the residents is married?

A farmer had hens and rabbits. These animals have 50 heads
and 140 feet. How many hens and rabbits does the farmer have?

 Objective
To be able to identify a function and to be
able to graphically represent functions.
 Purpose
To help describe input-output relations in
real-world applications and to use functions
to model and solve real-life problems.

Relation
 Relation – pairs of quantities that are related
to each other
 Example: The area A of a circle is related to
its radius r by the formula
.
2
r
A 


Function
 There are different kinds of relations.
 When a relation matches each item from one
set with exactly one item from a different set
the relation is called a function.

Definition of a Function
 A function is a relationship between two
variables such that each value of the first
variable is paired with exactly one value of
the second variable.
 The domain is the set of permitted x values.
 The range is the set of found values of y.
These can be called images.

Is it a Function?
 For each x, there is
only one value of y.
 Therefore, it IS a
function.
Domain, x Range, y
1 -3.6
2 -3.6
3 4.2
4 4.2
5 10.7
6 12.1
52 52

Is it a function?
 Three different y-
values (7, 8, and 10)
are paired with one x-
value.
 Therefore, it is NOT a
function
Domain, x Range, y
3 7
3 8
3 10
4 42
10 34
11 18
52 52

Function?
 Is it a function? State the domain and range.
 No. The x-value of 5 is paired with two
different y-values.
 Domain: (5, 6, 3, 4, 12)
 Range: (8, 7, -1, 2, 9, -2)
{(5, 8), (6, 7), (3, -1), (4, 2), (5, 9), (12, -2)

Vertical Line Test
 Used to determine if a graph is a function.
 If a vertical line intersects the graph at more
than one point, then the graph is NOT a
function.
NOT a Function

Is it a function? Give the domain and range.
 
 
4
,
4
:
2
,
4
:


Range
Domain
FUNCTION

Give the Domain and Range.
2
:
1
:


y
Range
x
Domain
3
0
:
2
2
:





y
Range
x
Domain

Functional Notation
 We have seen an equation written in the form
y = some expression in x.
 Another way of writing this is to use
functional notation.
 For Example, you could write y = x²
as f(x) = x².

Functional Notation: Find the following
( 3)
f 
2
( ) 3 2
f x x x
  
   
32
2
30
2
3
27
2
3
3
3
2







2
( ) 2
f x x x
  
3
( )
f m 
   
  
8
5
2
3
9
3
3
2
3
3
3
2
3
3
2
2
2

















m
m
m
m
m
m
m
m
m
m
m

Let’s look at Functions
Graphically

Find: 2 4
( ) ( )
f g

( )
f x ( )
g x

Find: 5 0
( ) ( )
f g

( )
f x ( )
g x

Find: 4 1
( ) ( )
f g
 
( )
f x ( )
g x

Find: 2 0
( ) ( )
f g
 
( )
f x ( )
g x

 A piecewise-defined function is a function that is
defined by two or more equations over a specified
domain.
 The absolute value function
can be written as a piecewise-defined function.
 The basic characteristics of the absolute value
function are summarized on the next page.
  x
x
f 

Absolute Value Function is a
Piecewise Function

Example
 Evaluate the function when x = -1 and 0.

 The domain of a function can be implied by
the expression used to define the function
 The implied domain is the set of all real
numbers for which the expression is defined.
 For example,

 The function has an implied
domain that consists of all real x other than
x = ±2
 The domain excludes x-values that result
in division by zero.

 Another common type of implied domain is
that used to avoid even roots of negative
numbers.
 EX:
is defined only for
The domain excludes x-values that result
in even roots of negative numbers.
.
0

x

 Objective:
To graph a function using domain and
range, even or odd, relative min/max.
 Purpose:
To introduce methods to help graph a
function.

Domain & Range of a Function
What is the
domain of
the graph of
the function
f?
 
4
,
1
: 
A

What is the
range of
the graph of
the function
f?
 
4
,
5


   .
2
1 f
and
f
Find 
  5
1 


f
  4
2 
f

 Let’s look at domain and range of a
function using an algebraic approach.
 Then, let’s check it with a graphical
approach.

Find the domain and range of
 Algebraic Approach
  .
4

 x
x
f
The expression under the radical can not be negative.
Therefore, Domain
.
0
4 

x
 


,
4
4
:
or
x
A Since the domain is never negative the
range is the set of all nonnegative real
numbers.
 


,
0
0
:
or
y
A
Range

Find the domain and range of
 Graphical Approach
  .
4

 x
x
f

Increasing and Decreasing
Functions

 The more you know about the graph of
a function, the more you know about
the function itself.
 Consider the graph on the next slide.

Falls from x = -2 to x = 0.
Is constant from x = 0 to
x = 2.
Rises from x = 2 to x = 4.

Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
Increases over
the entire real
line.

   



 ,
1
1
,
:
and
INCREASING
 
1
,
1
:

DECREASING

 
0
,
:


INCREASING
 
2
,
0
:
CONSTANT
 

,
2
:
DECREASING

Relative Minimum and
Maximum Values

Relative Min/Max
 The point at which a function changes
its increasing, decreasing, or constant
behavior are helpful in determining the
relative maximum or relative
minimum values of a function.

General Points – We’ll find
EXACT points later……

Approximating a Relative
Minimum
 Example: Use a GDC to approximate
the relative minimum of the function
given by
  .
2
4
3 2


 x
x
x
f

 Put the function into the “y = “ the
press zoom 6 to look at the graph.
 Press 2nd Calc, 3:minimum, left bound,
right bound, enter at the lowest point.
  .
2
4
3 2


 x
x
x
f

Example
 Use a GDC to approximate the relative
minimum and relative maximum of the
function given by
  .
3
x
x
x
f 



Solution
Relative Minimum
(-0.58, -0.38)

Solution
Relative Maximum
(0.58, 0.38)

Step Functions and
Piecewise-Defined Functions

Because of the vertical jumps, the greatest integer function is an example
of a step function.

Let’s graph a Piecewise-
Defined Function
 Sketch the graph of
 









1
,
4
1
,
3
2
x
x
x
x
x
f
Notice when open
dots and closed
dots are used. Why?

Algebraically
Let’s look at the graphs again and see if this applies.

Example
 Determine whether each function is
even, odd, or neither.

Algebraic
Graphical –
Symmetric to
Origin

Algebraic
Graphical –
Symmetric to y-
axis

Algebraic
Graphical – NOT
Symmetric to
origin OR y-axis.

You Try
 Is the function
 Even, Odd, of Neither?
  x
x
f 

Solution
  x
x
f 
Symmetric about the y-axis.

Lesson 2 - Functions and their Graphs - NOTES.ppt

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Lesson 2 - Functions and their Graphs - NOTES.ppt