1. Relations and Functions
Algebra II
Relations and Functions

2. VOCABULARY
1. For every input of a function there is exactly/at least/at most one output.

3. Relation (noun) ree LAY shun
VOCABULARY
Relation (noun) ree LAY shun
Related words: input, output, domain, range, function
Definition: A relation is a set of ordered pairs of input and output values.
Examples: {(0,1),(2,3),(0,4)}
𝒇 𝒙 = 𝒙 𝟐

4. RELATION
Input (domain)
{(0,1), (2,3), (0,4)}
Output (range)

5. Circle the output values in the relation.
{(1,0), (-2,3), (3,2), (5, -4)}

6. Write the domain of the relation.
{(-2,3), (-2,1), (5, 4)}
{−2,5}

7. There are 4 different ways to represent a relation:
Mapping diagram
Ordered pairs
Table
Graph

8. Representing a Relation
The average water temperature of the Gulf of Mexico in Key West, Florida varies during the year. In January, the average water temperature is 69 𝑜 F; in February, 70 𝑜 F; in March 75 𝑜 F; and in April, 78 𝑜 F. How can you represent this relation in four different ways?

9. Mapping Diagram Jan. Feb. Mar. Apr. 69 ____ 70 75 78
The average water temperature of the Gulf of Mexico in Key West, Florida varies during the year. In January, the average water temperature is 69 𝑜 F; in February, 70 𝑜 F; in March 75 𝑜 F; and in April, 78 𝑜 F. How can you represent this relation in four different ways?
Jan.
Feb.
Mar.
Apr. 69
____ 70 75 78

10. 70 75 78 {(Jan., 69), (Feb, ___), (Mar., ___), (Apr., ___)}
Ordered pairs
The average water temperature of the Gulf of Mexico in Key West, Florida varies during the year. In January, the average water temperature is 69 𝑜 F; in February, 70 𝑜 F; in March 75 𝑜 F; and in April, 78 𝑜 F. How can you represent this relation in four different ways? 70 75 78
{(Jan., 69), (Feb, ___), (Mar., ___), (Apr., ___)}

11. x (Month) y ( 𝒐 𝑭 ) Jan. 69 Feb. 70 Mar. 75 Apr. 78 Table
The average water temperature of the Gulf of Mexico in Key West, Florida varies during the year. In January, the average water temperature is 69 𝑜 F; in February, 70 𝑜 F; in March 75 𝑜 F; and in April, 78 𝑜 F. How can you represent this relation in four different ways?
x (Month)
y ( 𝒐 𝑭 )
Jan. 69
Feb. 70
Mar. 75
Apr. 78

12. The average water temperature of the Gulf of Mexico in Key West, Florida varies during the year. In January, the average water temperature is 69 𝑜 F; in February, 70 𝑜 F; in March 75 𝑜 F; and in April, 78 𝑜 F. How can you represent this relation in four different ways?
Graph

13. What is the domain and range of this relation?
{(-3, 14), (0, 7), (2, 0), (9, -18), (23, -99)}
First the set of x- coordinates:
Domain: {-3, 0, 2, 9, 23}
Range: {-99, -18, 0, 7, 14}
Then the set of y- coordinates:

14. The set of x-coordinates is the ________ of the relation.
domain
The set of x-coordinates is the ________ of the relation.
range
The set of y-coordinates is the ________ of the relation.

15. Identifying Functions
Circle the range values that correspond to domain value 2.
Domain
Range 2 3 4 6 -3 -1 3 6

16. True or False?
Every element of the input corresponds to exactly one element of the range.
TRUE

17. FALSE True or False? The relation is a function. 2 -3 3 -1 4 3 6 6
Domain
Range 2 3 4 6 -3 -1 3 6
FALSE

18. Vertical line test NO YES!! NO
A relation is a function if it can pass the vertical line test. Which means when the relation is on a graph you can not draw a straight vertical line and touch more than one point. NO
YES!! NO

19. Use the vertical line test. Which graph(s) represent(s) a function?NO
YES
YES

20. Underline the correct word(s) to complete the sentence.
If a vertical line passes through more than one point on a graph, the graph represents/does not represent a function.

21. For 𝑓 𝑥 =−4𝑥+1, what is 𝑓(−2)? Circle the given input. 2 −2 𝑥 −𝑥
Function Notation
For 𝑓 𝑥 =−4𝑥+1, what is 𝑓(−2)? Circle the given input. 2 −2 𝑥 −𝑥

22. Substitute the input into the function rule and simplify:
𝑓 𝑥 =−4𝑥+1; 𝑓(−2) 𝑓 =−4( )+1 = −2 −2
+8+1 =9

23. Writing and Evaluating a Function
You are buying bottles of a sports drink mix for a softball team. Each bottle costs $ What function rule models the total cost of a purchase? Evaluate the function for 15 bottles.

24. Writing and Evaluating a Function
You are buying bottles of a sports drink mix for a softball team. Each bottle costs $ What function rule models the total cost of a purchase? Evaluate the function for 15 bottles.
Let c = ?. Circle your choice.
cost per bottle cost of 15 bottles number of bottles bought total cost

25. Writing and Evaluating a Function
You are buying bottles of a sports drink mix for a softball team. Each bottle costs $ What function rule models the total cost of a purchase? Evaluate the function for 15 bottles.
Then 𝑓 𝑐 = ? Circle your choice.
cost per bottle cost of 15 bottles number of bottles bought total cost

26. Writing and Evaluating a Function
You are buying bottles of a sports drink mix for a softball team. Each bottle costs $ What function rule models the total cost of a purchase? Evaluate the function for 15 bottles.
Relate:
TOTAL COST is COST PER BOTTLE times NUMBER OF BOTTLES BOUGHT
Write:
𝑓 _______ =$___________ * _______ c
1.19 c

27. Writing and Evaluating a Function
You are buying bottles of a sports drink mix for a softball team. Each bottle costs $ What function rule models the total cost of a purchase? Evaluate the function for 15 bottles.
Evaluate at 15 bottles:
𝑓 𝑐 =1.19∗𝑐
𝑓 15 =1.19 ∗15
𝑓 15 =17.85
So it will cost $17.85 for 15 bottles.

28. Lesson Check
Error Analysis: Your friend writes, “In a function, every vertical line must intersect the graph in exactly one point.” Explain your best friends error and rewrite his statement so that it is correct.

29. Lesson Check
Draw a vertical line on each graph that does not intersect the function.

30. Lesson Check
Error Analysis: Your friend writes, “In a function, every vertical line must intersect the graph in exactly one point.” Explain your best friends error and rewrite his statement so that it is correct.
In a function, a vertical line must intersect the graph in exactly one point.

31. Math Success
Check off the vocabulary words you understand. □relation □function □domain □range Rate how well you can represent and interpret functions.
NEED TO REVIEW 2 4 6 8 10
NOW I GET IT!