### Linear Equations and Functions 4_5_4_6

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
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?
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.
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.
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  lw
  10 7   70
A  70
The area is 70 sq in
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
bh
height
2
Now, the height is 4 more than twice the base
h=4+2b
h  4  2 16   36
A
1
2
bh

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
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:
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.
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
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