2. 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.
3. 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?
4. 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?
5. A farmer had hens and rabbits. These animals have 50 heads
and 140 feet. How many hens and rabbits does the farmer have?
6. 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.
7. 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
8. 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.
9. 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.
10. 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
11. 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
12. 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)
13. 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
14. Is it a function? Give the domain and range.
4
,
4
:
2
,
4
:
Range
Domain
FUNCTION
15. Give the Domain and Range.
2
:
1
:
y
Range
x
Domain
3
0
:
2
2
:
y
Range
x
Domain
16. 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.
17. 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
24. 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
28. 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,
29. 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.
30. 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
32. Objective:
To graph a function using domain and
range, even or odd, relative min/max.
Purpose:
To introduce methods to help graph a
function.
33. Domain & Range of a Function
What is the
domain of
the graph of
the function
f?
4
,
1
:
A
34. Domain & Range of a Function
What is the
range of
the graph of
the function
f?
4
,
5
35. Domain & Range of a Function
.
2
1 f
and
f
Find
5
1
f
4
2
f
36. Let’s look at domain and range of a
function using an algebraic approach.
Then, let’s check it with a graphical
approach.
37. 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
38. Find the domain and range of
Graphical Approach
.
4
x
x
f
40. 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.
41. Falls from x = -2 to x = 0.
Is constant from x = 0 to
x = 2.
Rises from x = 2 to x = 4.
42. Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
Increases over
the entire real
line.
43. Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
,
1
1
,
:
and
INCREASING
1
,
1
:
DECREASING
44. Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
0
,
:
INCREASING
2
,
0
:
CONSTANT
,
2
:
DECREASING
46. 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.
48. 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
49. 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
50. Example
Use a GDC to approximate the relative
minimum and relative maximum of the
function given by
.
3
x
x
x
f
54. Because of the vertical jumps, the greatest integer function is an example
of a step function.
55. 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?
