College Mathematics: Learning Worksheets Chapter 2
35
10. A company keeps records of the total revenue (money taken in) in thousands of dollars
from the sale of x units (in thousands) of a product. It determines that total revenue is a
function R(x) given by
2
( ) 300 .
R
xxx=−
It also keeps records of the total cost of producing x units of the same product. It determines
that the total cost is a function C(x) given by
a) Find the break-even points for this company. (Round answer to nearest 1000.)
() ()
Rx Cx
b) Determine at what point profit is at a maximum. What is the maximum profit?
How many units must be sold in order to achieve maximum profit?
The profit equation is
() () ()
Px Rx Cx
=−
The maximum profit occurs at the vertex of the profit function. The x-coordinate is
the value into the function as follows:
2
( ) 260 1600
Px x x
36
11. The cost, C(x), of building a shed is a function of the number of square feet, x, in the
shed. If the cost function can be approximated by
2
( ) 0.01 20 25,000,Cx x x=−+ where 1000 3500x
a) What would be the cost of building a 1500-square-foot shed?
Substitute the value of 1500 into the cost function:
2
( ) 0.01 20 25,000
Cx x x
b) Find the minimum cost to build a shed. How many square-feet would that shed
have?
The minimum cost occurs at the vertex of the cost function. The x-coordinate is
2 2(0.01) 0.02
the value into the function as follows:
2
( ) 0.01 20 25000
Cx x x
37
12. The cost of producing computer software is a function of the number of hours worked by
the employees. If the cost function can be approximated by
2
( ) 0.04 20 6000,Cx x x=−+where 200 1000x
a) What would be the cost if the employees worked 800 hours?
Substitute the value of 800 into the cost function:
2
( ) 0.04 20 6000
Cx x x
b) Find the number of hours the employees should work in order to minimize the
cost. What would the minimum cost be?
The minimum cost occurs at the vertex of the cost function. The x-coordinate is
2 2(0.04) 0.08
the value into the function as follows:
2
( ) 0.04 20 6000
Cx x x
College Mathematics: Learning Worksheets Chapter 2
38
College Mathematics: Learning Worksheets Chapter 2
39
Name ________________________________ Date ______________ Class ____________
Goal: To describe and identify functions that are polynomial and rational in nature
For 1–6, determine each of the following for the polynomial functions:
a) the degree of the polynomial
b) the x-intercept(s) of the graph of the polynomial
c) the y-intercept of the graph of the polynomial
Section 2-4 Polynomial and Rational Functions
Definition: Polynomial function
1
110
() nn
nn
f
xaxax axa
for n a nonnegative integer, called the degree
of the polynomial. The coefficients 01
,,,
n
aa aare real numbers with 0.
n
a The
domain of a polynomial function is the set of all real numbers.
Definition: Rational function
()
() ()
nx
fx dx
() 0dx, where n(x) and d(x) are polynomials. The domain is the set
of all real numbers such that () 0.dx
Vertical Asymptotes:
Case 1: Suppose n(x) and d(x) have no real zero in common. If c is a real number such
that () 0,dxthen the line
x
cis a vertical asymptote of the graph.
Horizontal Asymptotes:
Case 1: If degree n(x) < degree d(x), then y = 0 is the horizontal asymptote.
Case 3: If degree n(x) > degree d(x), there is no horizontal asymptote.
College Mathematics: Learning Worksheets Chapter 2
40
1. 32
() 2 23 60 ( 3)( 5)( 4)fx x x x x x x=+ − −=+ − +
a) The degree of the polynomial is the highest exponent, which is 3.
b) The x-intercept(s) are found by setting the polynomial equal to zero. The
c) The y-intercept occurs when the x-value is zero. If the x-value is zero, the only
2. 32
() 8 972(3)(3)(8)fx x x x x x x=+ −−=− + +
a) The degree of the polynomial is the highest exponent, which is 3.
b) The x-intercept(s) are found by setting the polynomial equal to zero. The
c) The y-intercept occurs when the x-value is zero. If the x-value is zero, the only
3. 32
() 3 10 24 ( 3)( 2)( 4)fx x x x x x x
a) The degree of the polynomial is the highest exponent, which is 3.
b) The x-intercept(s) are found by setting the polynomial equal to zero. The
4. 32
() 4 4 ( 4)( 1)( 1)fx x x x x x x
a) The degree of the polynomial is the highest exponent, which is 3.
5. 432 2
() 2 2 2 ( 1)( 1)( 2 2)fxxxxx xxxx
a) The degree of the polynomial is the highest exponent, which is 4.
b) The x-intercept(s) are found by setting the polynomial equal to zero. The
6. 54 2 2
() 5 20 15 ( 3)( 1)( 1)( 2 5)fxxx xx x xxxx
a) The degree of the polynomial is the highest exponent, which is 5.
b) The x-intercept(s) are found by setting the polynomial equal to zero. The
College Mathematics: Learning Worksheets Chapter 2
42
For the given rational functions in 7–12,
a) Find the domain.
b) Find any x-intercept(s).
7. 1
() 2
fx
x
a) The function is defined everywhere except when the denominator is zero. The
domain is, therefore, all real numbers except 0.
b) The x-intercept(s) are found by setting the function equal to 0. Since the
function is a rational expression and the numerator cannot be zero, the
f)
43
8. 4
() 4
x
fx x
=
−
a) The function is defined everywhere except when the denominator is zero.
4
x
=
The domain is, therefore, all real numbers except 4.
b) The x-intercept(s) are found by setting the function equal to 0. Since the
function is a rational expression, the function value is zero when the
numerator is zero.
c) The y-intercept is found when the value of x is zero.
4
() 4
x
fx x
=
−
d) Vertical asymptotes occur at values where the function is not defined. Since
the function is not defined for x = 4, the vertical asymptote is the line x = 4.
e) Horizontal asymptotes are found by dividing all terms by the highest power of
4
44
x
x
x
f(x) tends to 4. The horizontal asymptote is the line y = 4.
f)
44
9. 5
() 2
x
fx x
=
−
a) The function is defined everywhere except when the denominator is zero.
b) The x-intercept(s) are found by setting the function equal to 0. Since the
function is a rational expression, the function value is zero when the
numerator is zero.
c) The y-intercept is found when the value of x is zero.
5
() 2
x
fx x
=
−
d) Vertical asymptotes occur at values where the function is not defined. Since
e) Horizontal asymptotes are found by dividing all terms by the highest power of
5
55
x
x
x
f))
45
10. 24
() 3
x
fx x
a) The function is defined everywhere except when the denominator is zero.
b) The x-intercept(s) are found by setting the function equal to 0. Since the
function is a rational expression, the function value is zero when the
numerator is zero.
c) The y-intercept is found when the value of x is zero.
24
() 3
x
fx x
3
d) Vertical asymptotes occur at values where the function is not defined. Since
the function is not defined for x = –3, the vertical asymptote is the line x = –3.
e) Horizontal asymptotes are found by dividing all terms by the highest power of
x
244
2
24
x
x
xx
x
f)
11. 4
() 4
x
fx
x
a) The function is defined everywhere except when the denominator is zero.
40
x
x
b) The x-intercept(s) are found by setting the function equal to 0. Since the
function is a rational expression, the function value is zero when the
numerator is zero.
c) The y-intercept is found when the value of x is zero.
4
() 4
x
fx
x
d) Vertical asymptotes occur at values where the function is not defined. Since
e) Horizontal asymptotes are found by dividing all terms by the highest power of
44
1
4
x
xx x
x
f)
47
12. 15
() 12
x
fx
x
a) The function is defined everywhere except when the denominator is zero.
12 0
x
2
b) The x-intercept(s) are found by setting the function equal to 0. Since the
function is a rational expression, when the numerator is zero, the function
value is zero.
15 0
x
5
c) The y-intercept is found when the value of x is zero.
15
() 12
x
fx
x
d) Vertical asymptotes occur at values where the function is not defined. Since
e) Horizontal asymptotes are found by dividing all terms by the highest power of
5
11
5
15
x
x
f)
13. A video production company is planning to produce a documentary. The producer
estimates that it will cost $104,000 to produce the video and $40 per video to copy and
distribute the tape.
a) Assuming that the total cost to market the video, C(n), is linearly related to the
b) The average cost per video for an output of n videos is given by ()
() Cn
Cn n
.
+
c) Sketch a graph of the average cost function for 1000 6000n≤≤ .
d) What does the average cost per video tend to as production increases?
To find the value that the function tends to go towards, you will find the horizontal
asymptote of the function. Horizontal asymptotes are found by dividing all terms by the
104,000 104,000
40 40
n
nn n
++
49
14. A contractor purchases a piece of equipment for $36,000. The equipment requires an
average expenditure of $8.25 per hour for fuel and maintenance, and the operator is paid
$13.50 per hour to operate the machinery.
a) Assuming that the total cost per day, C(h), is linearly related to the number of
hours, h, that the machine is operated, write an equation for the cost function.
b) The average cost per hour of operating the machine is given by ()
() Ch
Ch h
.
c) Sketch a graph of the average cost function for 1000 8000h .
d) What cost per hour does the average cost per hour tend to as the number of hours
of use increases?
To find the value that the function tends to go towards, you will find the horizontal
asymptote of the function. Horizontal asymptotes are found by dividing all terms by
36,000 36,000
21.75 21.75
n
hh h
15. The daily cost function for producing x printers for home computers was
determined to be
2
( ) 8 6000,Cx x x=++
The average cost per printer at a production level of x printers per day is ()
() Cx
Cx
x
.
a) Find the average cost function.
b) Sketch a graph of the average cost function for 25 150x .
c) At what production level is the daily average cost at a minimum? What is that
minimum value?
(75) 163
(76) 162.947
C
C
Therefore, the minimum average cost of 162.92 occurs when 77 printers are
16. The monthly cost function for producing x brake assemblies for a certain type of car is
given by
The average cost per brake assembly at a production level of x assemblies per month is
()
() Cx
Cx
x
.
a) Find the average cost function.
b) Sketch a graph of the average cost function for 0 150x .
The x-axis scale shown is from 0 to 150. Each tick mark is 25 units. The y-axis scale
shown is from 300 to 500. Each tick mark is 50 units.
c) At what production level is the daily average cost at a minimum? What is that
minimum value?
(54) 364.667
(56) 364.714
C
C