1. Functional Relationships

2. Functional Relationships
Day 1

3. 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).
f(x) y x

4. Sketch a linear function. Sketch a nonlinear function.
Makes a line
Non-Linear Function:
Does not make a line.
Sketch a linear function. Sketch a nonlinear function.

5. 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 + 2
The range is the y or output value in a function.
(set of 2nd coordinates of the ordered pairs)
A relation is a set of ordered pairs.
{(3, 2), (4, 2), (-2, 1)}

6. Given the relation {(3,2), (1,6), (-2,0)}, find the domain and range.

7. The relation {(2,1), (-1,3), (0,4)} can be shown byx y
1) a table.
2) a mapping.
3) a graph. 2 -1 1 3 4 2 -1 1 3 4

8. 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

9. You do not need to write 3 twice in the range!
Mapping x y -1 4 7 3 6 -1
You do not need to write 3 twice in the range!

10. What is the domain of the relation {(2,1), (4,2), (3,3), (4,1)}
{2, 3, 4, 4}
{1, 2, 3, 1}
{2, 3, 4}
{1, 2, 3}
{1, 2, 3, 4}
Answer Now

11. What is the range of the relation {(2,1), (4,2), (3,3), (4,1)}
{2, 3, 4, 4}
{1, 2, 3, 1}
{2, 3, 4}
{1, 2, 3}
{1, 2, 3, 4}
Answer Now

12. 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!

13. Vertical Line Test
FUNCTION!
NO!
NO WAY!
FUNCTION!

14. Relation = {(-1,3), (0,6), (4,-1), (7,3)}
Given the following table, show the relation, domain, range, and mapping. x y
Relation = {(-1,3), (0,6), (4,-1), (7,3)}
Domain = {-1, 0, 4, 7}
Range = {3, 6, -1, 3}

15. 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

16. Functional Relationships
Day 2

17. 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: ≤ x ≤ 8
RANGE
DOMAIN
y values: -3 through +8
which can be written as: ≤ y ≤ 8

18. 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: ≤ x ≤ 6
RANGE
DOMAIN
y values: -4 through +7
which can be written as: ≤ y ≤ 7
IS THIS A FUNCTION??

19. Functional Relationships
Day 3

20. Relations & Functions-YEAR 1
A function is like a machine. You put something in and you get something out.
Input x
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.
Rule
f(x)
Output

21. 2x Function Notation 10 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 5 2x 10
2(5) 10
Output

22. EXAMPLE: Complete the table to find out the human ages of dogs ages 3 through 6.
INPUT
Human Years
RULE
OUTPUT
Dog years x 7x
f(x)
7(3) 3 21 4
7(4) 28 5
7(5) 35 6
7(6) 42
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 …

23. EXAMPLE: Make a function table to find the range of f(x) = 3x + 5 if the domain is {-2, -1, 0, 3, 5}. x
3x + 5
f(x) -2
3(-2) + 5 -1 -1
3(-1) + 5 2
3(0) + 5 5 3
3(3) + 5 14 5
3(5) + 5 20
Range: {-1, 2, 5, 14, 20}.

24. More Examples EXAMPLE: Find f(-3) if f(n) = -2n – 4 f(-3)= -2(-3) – 4