1.5 i understand functions. function for each input there is only one output ex: let g represent the...
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1.5 I understand Functions
Function
• For each input there is only one output• Ex: Let g represent the age of a US citizen and
d represent the number of days since his or her birth. Is g a function of d or is d a function of g?
If y is a function of x
• Independent variable: x • Dependent variable: y
• The DETERMINES the value of .
Relation
• A set of ordered pairs• Can be indicated by a verbal description, a
graph, a formula or equation, or a table• Not every relation is a function, but every
function is a relation.
Determine if each relation is a function:
•
X Y
1 9
20 29
6 10
-1 9
X Y
5 7
6 8
7 9
5 6
Vertical Line Test
• A graph is a function if and only if there is no vertical line that crosses the graph more than once
• AKA: When you draw a vertical line on the graph, if it intersects with the graph more than once, it is NOT a function
Use the Vertical Line Test to determine if the graph is a function
Identifying a function from an equation
•
Domain and Range
•
Determine the Domain and Range (Put into calculators if needed)
•
12 xy
Function Notation
• A function defined by a set of ordered pairs can be named with a letter
= { (2,5), (3,8)}Since pairs 2 with 5, we write f(2)=5The value at 2 is 5.f(3)=8; The value at 3 is 8.
Using Function Notation
• 3)( xxf
Using Function Notation with Variables
• 2)( 2 xxf
The Average Rate of Change
• Slope =
• It is not necessary to have a formula for a function to find the average rate of change, you can find the average rate of change (slope) if you know two points on the graph.
12
12
xx
yy
Average Rate of Change (slope)
• Find the average rate of change between the two points.
Example 1
• The population of California was 29.8 million in 1990 and 33.9 million in 2000. What was the average rate of change of the population over that time interval?
• Use the coordiantes (year, population) (1990, 29,800,000) and (2000, 33,900,000)
The Difference Quotient
h
xfhxf )()(
Find and simplify the difference quotient
•
Find and simplify the difference quotient
•
Find and simplify the difference quotient
xxf )(
1.6 Graphs of Relations and Functions
• Parent Functions• Graph the following parent functions and
state the domain and range.
X Y
X Y
X Y
The Absolute Value Function
The square root function:
xy
xy
X Y
The Cubic function:
X Y
3xy
X Y
X Y
X Y
X Y