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Writing a Function Rule. Formalizing Relations and Functions 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 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? 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 W=50,000+420m 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 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. Answer C=12+15(10) C=12+15n C=12+150=162 $162 And a 5 day is C=12+15(5) C=12+75= $87 5 days stay is 150/2=75 No, making the stay shorter only halves the daily charge, not the bath charge. 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 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.? Width(w) Length(l=w+3) w 3 7 3 10 A lw 10 7 70 A 70 The area is 70 sq in 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. A 1 bh height 2 Now, the height is 4 more than twice the base h=4+2b h 4 2 16 36 A 1 2 bh 1 2 16 36 288 base Can you graph the function rule from the problem. How do you know the rule is nonlinear? Can we graph these? A 2b b 2 1 Area, A 2 3 4 5 6 7 8 9 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 0 1 2 3 base, b The graph is not a line but it is a function. 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 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)} Domain={-2,0,4,5} Range={0.5,2.5,6.5} B.{(6,5),(4,3),(6,4)(5,8)} Domain={4,5,6} Range={3,4,5,8} -2 0 4 5 0.5 2.5 6.5 It is a function 4 5 6 It is not a function 3 4 5 8 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)} Not a function B.{(-1,1),(-2,2),(4,-4),(7,-7)} function -1 -2 4 7 1 2 -4 -7 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: Your turn function 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. Answer 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 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? 455 words 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} D 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? 4 .f(x)= 2x and g(x)=x 1.Find the value a)f(3)+g(4) 23 b)f(5)-2g(1) 6 2