Upload
maude-cunningham
View
217
Download
0
Embed Size (px)
Citation preview
Functional Relationships
Functional Relationships
Day 1
Vocabulary:
A function is a relation in which each element of the domain is paired with exactly one element of the range. Another way of saying it is that there is one and only one output (y) per input (x).
x yf(x)
Sketch a linear function. Sketch a nonlinear function.
Linear Function:
Makes a line
Non-Linear Function:
Does not make a line.
How about some more definitions?
The domain is the x or input value in a function.(set of 1st coordinates of the ordered pairs)
(2, 0) or y = 3x + 2The range is the y or output value in a function.
(set of 2nd coordinates of the ordered pairs)
(2, 0) or y = 3x + 2
A relation is a set of ordered pairs.{(3, 2), (4, 2), (-2, 1)}
Given the relation {(3,2), (1,6), (-2,0)}, find the domain and range.
Domain = {3, 1, -2}
Range = {2, 6, 0}
The relation {(2,1), (-1,3), (0,4)} can be shown by
1) a table.
2) a mapping.
3) a graph.
x y2-10
134
2-10
134
How can you tell if a relation is a function without a graph? Only ONE output per input Coordinates: Check all x values. X’s can not
be repeated Mapping: Can only have one line drawn from
each x Graph: passes vertical line test
Mappingx -1 0 4 7y 3 6 -1 3
You do not need to write 3 twice in the range!
-1047
36-1
What is the domain of the relation{(2,1), (4,2), (3,3), (4,1)}
1. {2, 3, 4, 4}
2. {1, 2, 3, 1}
3. {2, 3, 4}
4. {1, 2, 3}
5. {1, 2, 3, 4}
Answer Now
What is the range of the relation{(2,1), (4,2), (3,3), (4,1)}
1. {2, 3, 4, 4}
2. {1, 2, 3, 1}
3. {2, 3, 4}
4. {1, 2, 3}
5. {1, 2, 3, 4}
Answer Now
Vertical Line Test (pencil test)
If any vertical line passes through more than one point of the graph, then that relation is not
a function.
Are these functions?
FUNCTION! FUNCTION! NOPE!
Vertical Line Test
NO WAY! FUNCTION!
FUNCTION!
NO!
Given the following table, show the relation, domain, range, and mapping.
x -1 0 4 7y 3 6 -1 3
Relation = {(-1,3), (0,6), (4,-1), (7,3)}
Domain = {-1, 0, 4, 7}
Range = {3, 6, -1, 3}
Other Related Vocabulary:
Independent Variable (input): the variable that determines the value of the
dependent variable. (x axis or domain values)
Dependent Variable (output): The variable relying on the independent variable (y
axis or range values)
EXAMPLE: the diameter of a pizza and its cost
Functional Relationships
Day 2
Finding Domain and Range of a Graph
First identify all possible values for the domain (x or input).
Next, identify all possible values for the range (y or output).
x values: -9 through +8
which can be written as: -9 ≤ x ≤ 8
y values: -3 through +8
which can be written as: -3 ≤ y ≤ 8
DOMAIN
RA
NG
E
Practice: Finding the Domain and Range of a Graph First identify all
possible values for the domain (x or input).
Next, identify all possible values for the range (y or output).
x values: -5 through +6
which can be written as: -5 ≤ x ≤ 6
y values: -4 through +7
which can be written as: -4 ≤ y ≤ 7
DOMAIN
RA
NG
E
IS THIS A FUNCTION??
Functional Relationships
Day 3
Relations & Functions-YEAR 1
A function is like a machine. You put something in and you get something out.
Sometimes equations have two variables. When there are two variables in the equation, all solutions are ordered pairs. (x, f(x))
There are an infinite number of solutions for a two variable equation.
Input
Output
Rule
f(x)
x
Function Notation
For example, with a function f(x) = 2x,
if the input is 5, then it is written as
f(5) = 2(5)
The output is ____.
Input
Output
2x
5
2(5)
10
EXAMPLE: Complete the table to find out the
human ages of dogs ages 3 through 6.
So, a 3 year old dog is 21 in human years … 4 year old dog is 28 … … 5 year old dog is 35 … … 6 year old dog is 42 …
INPUTHuman Years
RULE OUTPUTDog years
x 7x f(x)
7(6)
7(5)
7(4)
7(3)
426
355
284
213
EXAMPLE: Make a function table to find the range of
f(x) = 3x + 5 if the domain is {-2, -1, 0, 3, 5}.
3(5) + 5
3(3) + 5
3(0) + 5
3(-1) + 5
3(-2) + 5
3x + 5
205
143
50
2-1
-1-2
f(x)x
Range: {-1, 2, 5, 14, 20}.
More Examples
EXAMPLE: Find f(-3) if f(n) = -2n – 4
EXAMPLE: Find nnff 315)( if 3
1
f(-3)= -2(-3) – 4 f(-3) = 2
3
1315
3
1f
143
1
f