C. Graphs of Basic Power Functions
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Section 2.4 Basic Rational Functions and Power Functions; More on the Domain
1
3
5
The functions
y ϭ x2, y ϭ x4, y ϭ x3, y ϭ 1
x, and y ϭ x2 are all power functions,
1
5
4
but only y ϭ x and y ϭ 1 x are also root functions. Initially we will focus on
power functions where p 7 0.
EXAMPLE 5
ᮣ
Comparing the Graphs of Power Functions
Use a graphing calculator
to graph the power functions f 1x2 ϭ x4, g1x2 ϭ x3,
3
1
2
h1x2 ϭ x , p1x2 ϭ x , and q1x2 ϭ x2 in the standard viewing window. Make an
observation in QI regarding the effect
of the 7exponent on each function, then
1
discuss what the graphs of y ϭ x6 and y ϭ x2 would look like.
1
Solution
ᮣ
First we enter the functions in sequence
as Y1 through Y5 on the Y= screen
(Figure 2.61). Using ZOOM 6:ZStandard
produces the graphs shown in
Figure 2.62. Narrowing the window
to focus on QI (Figure 2.63:
x ʦ 3 Ϫ4, 104, y ʦ 3Ϫ4, 10 4 ), we
quickly see that for x Ն 1, larger
values of p cause the graph of y ϭ x p
to increase at a faster rate, and smaller
values at a slower rate. In other words
1
1
(for x Ն 1), since 6 , the graph of
6
4
1
y ϭ x6 would increase slower and appear
1
to be “under” the graph of Y1 ϭ X4.
7
7
Since 7 2, the graph of y ϭ x2 would
2
increase faster and appear to be “more
narrow” than the graph of Y5 ϭ X2
(verify this).
2
Figure 2.61, 2.62
10
Ϫ10
10
Ϫ10
Figure 2.63
10 Y5
Y4
Y3
Y2
Y1
Ϫ4
10
Ϫ4
Now try Exercises 39 through 48
ᮣ
The Domain of a Power Function
In addition to the observations made in Example 5, we can make other important notes,
particularly regarding the domains of power functions. When the exponent on a power
m
7 0 in simplest form, it appears the domain is all real
function is a rational number
n
2
1
numbers if n 2 is odd, as seen in the graphs of g1x2 ϭ x3 , h1x2 ϭ x1 ϭ x1 , and
2
q1x2 ϭ x ϭ x1. If n is an even1 number, the domain
is all nonnegative real numbers as
3
seen in the graphs of f 1x2 ϭ x4 and p1x2 ϭ x2. Further exploration will show that if p is
irrational, as in y ϭ x, the domain is also all nonnegative real numbers and we have
the following:
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CHAPTER 2 More on Functions
The Domain of a Power Function
Given a power function f 1x2 ϭ x p with p 7 0.
m
1. If p ϭ is a rational number in simplest form,
n
a. the domain of f is all real numbers if n is odd: x ʦ 1Ϫq, q 2 ,
b. the domain of f is all nonnegative real numbers if n is even: x ʦ 3 0, q 2 .
2. If p is an irrational number, the domain of f is all nonnegative real numbers:
x ʦ 30, q 2 .
Further confirmation
of statement
1 can be found by recalling the graphs of
1
1
3
y ϭ 1x ϭ x2 and y ϭ 1
x ϭ x3 from Section 2.2 (Figures 2.64 and 2.65).
Figure 2.64
y
Figure 2.65
f(x) ϭ ͙x
5
5
(note n is even)
(9, 3)
(4, 2) (6, 2.4)
(0, 0)
Ϫ1
(note n is odd)
(8, 2)
(1, 1)
(0, 0)
4
8
3
y g(x) ϭ ͙x
x
Ϫ8
Ϫ4
(1, 1)
4
8
x
(Ϫ1, Ϫ1)
(Ϫ8, Ϫ2)
Ϫ5
Ϫ5
Domain: x ʦ 30, q 2
Range: y ʦ 3 0, q 2
EXAMPLE 6
ᮣ
Domain: x ʦ 3Ϫq, q2
Range: y ʦ 3Ϫq, q2
Determining the Domains of Power Functions
State the domain of the following power functions, and identity whether each is
also a root function.
4
1
2
8
a. f 1x2 ϭ x5 b. g1x2 ϭ x10 c. h1x2 ϭ 1
x d. q1x2 ϭ x3 e. r 1x2 ϭ x1 5
Solution
ᮣ
a. Since n is odd, the domain of f is all real numbers; f is not a root function.
b. Since n is even, the domain of g is x ʦ ΄0, q 2 ; g is a root function.
1
c. In exponential form h1x2 ϭ x8. Since n is even, the domain of h is x ʦ ΄0, q 2 ;
h is a root function.
d. Since n is odd, the domain of q is all real numbers; q is not a root function
e. Since p is irrational, the domain of r is x ʦ ΄0, q 2 ; r is not a root function
Now try Exercises 49 through 58
ᮣ
Transformations of Power and Root Functions
As we saw in Section 2.2 (Toolbox Functions and Transformations), the graphs of the
3
root functions y ϭ 1x and y ϭ 1 x can be transformed using shifts, stretches,
reflections, and so on. In Example 8(b) (Section 2.2) we noted the graph of
3
3
h1x2 ϭ 2 1
x Ϫ 2 Ϫ 1 was the graph of y ϭ 1
x shifted 2 units right, stretched by a
factor of 2, and shifted 1 unit down. Graphs of other power functions can be transformed in exactly the same way.
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Section 2.4 Basic Rational Functions and Power Functions; More on the Domain
EXAMPLE 7
ᮣ
Graphing Transformations of Power Functions
Based on our previous observations,
2
3
a. Determine the domain of f 1x2 ϭ x3 and g1x2 ϭ x2 , then verify by graphing
them on a graphing calculator.
2
3
b. Next, discuss what the graphs of F1x2 ϭ 1x Ϫ 22 3 Ϫ 3 and G1x2 ϭ Ϫx2 ϩ 2
will look like, then graph each on a graphing calculator to verify.
Solution
ᮣ
m
a. Both f and g are power functions of the form y ϭ x n . For f, n is odd so its
domain is all real numbers. For g, n is even and the domain is x ʦ 30, q2 .
Their graphs support this conclusion (Figures 2.66 and 2.67).
Figure 2.66
Figure 2.67
10
Ϫ10
10
10
Ϫ10
Ϫ10
10
Ϫ10
b. The graph of F will be the same as the graph of f, but shifted two units right
and three units down, moving the vertex to 12, Ϫ32 . The graph of G will be the
same as the graph of g, but reflected across the x-axis, and shifted 2 units up
(Figures 2.68 and 2.69).
Figure 2.69
Figure 2.68
10
Ϫ10
C. You’ve just seen how
we can graph basic power
functions and state their
domains
10
10
Ϫ10
Ϫ10
10
Ϫ10
Now try Exercises 59 through 62
ᮣ
D. Applications of Rational and Power Functions
These new functions have a variety of interesting and significant applications in the
real world. Examples 8 through 10 provide a small sample, and there are a number of
additional applications in the Exercise Set. In many applications, the coefficients may
be rather large, and the axes should be scaled accordingly.
EXAMPLE 8
ᮣ
Modeling the Cost to Remove Waste
For a large urban-centered county, the cost to remove chemical waste and other
Ϫ18,000
Ϫ 180,
pollutants from a local river is given by the function C1p2 ϭ
p Ϫ 100
where C( p) represents the cost (in thousands of dollars) to remove p percent of
the pollutants.
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a. Find the cost to remove 25%, 50%, and 75% of the pollutants and comment on
the results.
b. Graph the function using an appropriate scale.
c. Use mathematical notation to state what happens as the county attempts to
remove 100% of the pollutants.
Solution
ᮣ
a. We evaluate the function as indicated, finding that C1252 ϭ 60, C1502 ϭ 180,
and C1752 ϭ 540. The cost is escalating rapidly. The change from 25% to 50%
brought a $120,000 increase, but the change from 50% to 75% brought a
$360,000 increase!
C(p)
b. From the context, we need only graph the
x ϭ 100
1200
portion from 0 Յ p 6 100. For the C-intercept
we substitute p ϭ 0 and find C102 ϭ 0, which
900
seems reasonable as 0% would be removed
(75, 540)
600
if $0 were spent. We also note there must be
a vertical asymptote at x ϭ 100, since this
300
x-value causes a denominator of 0. Using
(25, 60)
(50, 180)
p
this information and the points from part (a)
100
50
75
25
produces the graph shown.
c. As the percentage of pollutants removed
y ϭ Ϫ180
approaches 100%, the cost of the cleanup
skyrockets. Using notation: as p S 100 Ϫ , C S q .
Now try Exercises 65 through 70
ᮣ
While not obvious at first, the function C(p) in Example 8 is from the family of
1
reciprocal functions y ϭ . A closer inspection shows it has the form
x
Ϫ18,000
1
Ϫa
Ϫk S
Ϫ 180, showing the graph of y ϭ is shifted right
yϭ
x
xϪh
x Ϫ 100
100 units, reflected across the x-axis, stretched by a factor of 18,000 and shifted 180 units
down (the horizontal asymptote is y ϭ Ϫ180). As sometimes occurs in real-world
applications, portions of the graph were ignored due to the context. To see the full
graph, we reason that the second branch occurs on the opposite side of the vertical and
horizontal asymptotes, and set the window as shown in Figure 2.70. After entering
C(p) as Y1 on the Y= screen and pressing GRAPH , the full graph appears as shown in
Figure 2.71 (for effect, the vertical and horizontal asymptotes were drawn separately
using the 2nd PRGM (DRAW) options).
Figure 2.71
Figure 2.70
2000
200
0
Ϫ2000
Next, we’ll use a root function to model the distance to the horizon from a
given height.