1.2 Domain and Range
Define:
Domain
Range
Example of reasonable domain:
f(age) = average height. What is a reasonable domain and range?
Introduce set-builder and interval notation.
Example: Write
and
Example: Write
as interval notation
and
in set builder notation
Domain and range from a graph
Example:
2.5
2
1.5
1
0.5
0
-4
-4
-3
-3
-2
-2
-1
4
3
2
1
0
-1 -1 0
-2
-3
-4
-5
-6
-7
0
1
1
2
2
3
3
4
Review domain and range of the toolkit functions: Constant, Identity (or Linear), Absolute Value,
Quadratic, Cubic, Square Root, Cube Root, Reciprocal (or Rational), and Reciprocal Squared.
Introduce piecewise functions.
Example: write a piecewise function to describe a cell phone plane with a base fee of $30 and includes
300 free text messages, then 10 cents for each additional text.
{
Answer:
Example: evaluate
,
,
based on
{
Example: Write an equation for
5
4
3
2
1
0
-3
Answer:
-2
-1
0
1
2
3
4
{
1.3 Rates of Change and Behavior of Graphs
Examples of units for rates of change: dollars per hour, miles per hour, pounds per square inch, people
per year, etc.
Average Rate of Change =
Calculate average rate of change from a table:
Example: estimate the rate of change of
2 4 6 8 10
3 6 10 16 26
on the intervals [4, 8] and [4, 10]
Calculate average rate of change from a graph
10
9
8
7
6
5
4
3
2
1
0
-4
Example
-3
-2
-1
0
1
2
3
4
on the intervals [0,1] and [1,2] and [0, 2]
Calculate average rate of change from a function over several intervals
Example: Given
find the average rate of change on [-1, 3], [1, 3], [2, a], [3, 3+h], [a, a+h]
Define increasing/decreasing and local extrema.
Increasing: The function values increase as the inputs increase (average rate of change is positive)
Decreasing: The function values decrease as the inputs increase (average rate of change is negative)
Local Extrema (Two Types)
Local Maximum: Point where a function changes from increasing to decreasing
Local Minimum: Point where a function changes from decreasing to increasing
Determine intervals of increasing/decreasing and extrema from a calculator drawn graph.
Example:
Estimate intervals of increasing/decreasing from a table
t
7
8
9
10
11
12
13
P(t)
0.97 1.2
1.75 2.01 1.9
1.8
1.81
Define concavity
Concave up: rate of change is increasing (holds water, smiles)
Concave down: rate of change is decreasing (spills water, frowns)
A point where the concavity changes is called the inflection point
30
20
10
0
-4
-3
-2
-1
-10
0
1
2
3
4
-20
-30
Estimate intervals of concavity from graph.
20
15
10
5
0
-3
-2
-1
-5
-10
-15
0
1
2
3
4