1. Writing a Function Rule. Formalizing Relations and Functions What you’ll learn To write equations that represent functions. To determine whether a relation is a function. To find domain and range and use of the function notation Relation: often represented as a set of ordered pairs (x,y) Domain: is the set of x values Range : is the set of the y values Vertical line test: is a test to find out is a relation is a function

2. Writing a Function Rule. A landfill has 50,000 tons of waste in it. Each month it accumulates an average of 420 more tons of waste What equation is a function rule that represents the total amount of waste after m months? W=50,000+420m Let’s W be total amount of waste And we know m is the months 50,000 tons are already there and each month it is accumulating 420 more

3. Your turn Carolyn has 420 CDs in her collection. Each month, she Adds 12 more CDs to her collection. What equation is a function rule that represents this situation? Answer: C=420+12m

4. Writing and Evaluating a Function Rule. A kennel charges $15 per day to board dogs. Upon arrival, each dog must have a flea bath that cost $12. Write a function rule for the total cost for n days of boarding plus a bath. How much does a 10-day stay cost? And a 5 day stay cost half as much as a 10 day stay? Explain. C=12+15n $162 C=12+15(10) C=12+150=162 And a 5 day is C=12+15(5) C=12+75= $87 No, making the stay shorter only halves the daily charge, not the bath charge. 5 days stay is 150/2=75 Answer

5. Your turn An archery club charges an annual membership fee of $65 plus $2 per visit. Write a function rule for the total cost of belonging to the club if you make v visits in a year. How much would it cost if you use the club 15 times in a year? C=65+2v C=65+2(15)=95 It will cost $95.00

6. Writing a Nonlinear Function Rule Write a function rule for the area of a rectangle whose length is 3 in. more than the width. What is the area of the rectangle when its width is 7 in.? Length(l=w+3) Width(w) The area is 70 sq in

7. Your turn Write a function rule for the area of a triangle whose height is 4 in. more than twice the length of its base. What is the area of the triangle when the length of its base is 16 in. base height Now, the height is 4 more than twice the base h=4+2b

8. Can you graph the function rule from the problem. How do you know the rule is nonlinear? b=1 then A=3 b=2 then A=8 b=3 then A=15 b=0 then A=0 b=-1 then A=-1 b=-2 then A=0 The graph is not a line but it is a function. Can we graph these? Area, A 1 2 3 4 5 6 7 8 9 0 1 2 3 base, b

9. Formalizing relations and Functions Take a note: A relation is a pairing of numbers in one set, it is often represented as a set of ordered pairs (x,y). In this case the set of the x-values is called Domain and the set of y-values is called Range. Is very important to know that a function is a special type of relation in which each value in the domain is paired with EXACTLY one value in the range

10. Problem 1: Identify the domain and the range of each relation Represent the relation with a mapping diagram. Is this relation a function? A.{(-2,0.5), (0,2.5),(4,6.5),(5,2.5)} B.{(6,5),(4,3),(6,4)(5,8)} Domain={-2,0,4,5} Range={0.5,2.5,6.5} Domain={4,5,6} Range={3,4,5,8} -2 0 4 5 0.5 2.5 6.5 456456 34583458 It is a function It is not a function

11. Your turn Identify the domain and range of each relation. Represent the relation with a mapping diagram. Is the relation a function? A.{(4.2,1.5),(5,2.2),(7,4.8),(4.2,0)} B.{(-1,1),(-2,2),(4,-4),(7,-7)} Not a function -2 4 7 1 2 -4 -7 function

12. Just a note: Another way to find if a relation is a function is using the vertical line test. If any vertical line passes through more than one point of the graph, then for some domain value there is more than one range value. So the relation is not a function. Examples:

13. Your turn function

14. Take a note: There is another way to write functions and it is called function notation. f(x)=3x+2 Identifying a Reasonable Domain and Range A charter boat travels at a maximum rate of 25 miles per hour. The function d(x)=25x represents the distance d(x), in miles, that the boat can travel in x hours. The charter boat travels a maximum distance of 75 miles from the shore. Domain all real numbers ≥0 and ≤ 3 Range: all real numbers ≥ 0 and ≤ 75 d(0)=25(0)=0 d(1)=25(1)=25 d(2)=25(2)=50 d(3)=25(3)=75 Answer

15. Your turn 1.The function T (x) =65x represents the number of words T(x) that Rachel can type in x minutes. How many words can she type in 7 minutes? 2. What is the range of f (x) =3x - 2 with domain {1, 2, 3, 4}? A. {3, 6, 9, 12} B. {21, 2, 5, 8} C. {2, 4, 6, 8} D. {1, 4, 7, 10 } 3. Lorena has 4 rolls of ribbon to make party favors. Each roll can be used to make 30 favors. The function F (r) = 30r represents the number of favors F(r) that can be made with r rolls. What are a reasonable domain and range of the function? What is a graph of the function? 455 words 4.f(x)= 2x and g(x)=.Find the value a)f(3)+g(4) b)f(5)-2g(1) 236 D