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Power Functions with Modeling Worksheet

Power Functions with Modeling Worksheet
  • Page 1
 1.  
If f(x) = xn , n is a positive integer is an odd function, then which of the following is correct for the function g(x) = xn+3.
a.
g(x) is an even function
b.
g(x) is neither even nor odd
c.
g(x) is an odd function
d.
None of the above


Solution:

f(x) = xn, n is a positive integer is an odd function.

So, f(- x) = - f(x)
[Definition of an odd function.]

(- x)n = - xn

(- 1)n xn = - xn
[Divide both sides by xn.]

(- 1)n = - 1

n = 1, or 3, or 5, . . .
[n is a positive integer.]

So, n + 3 = 1 + 3, or 3 + 3, or 5 + 3, . . .

= 4, or 6, or 8, . . . and hence n + 3 is an even number.

g(- x) = (- x)n + 3
[Replace x by - x is g(x) = xn + 3.]

= (- 1) n + 3 xn + 3

= xn + 3 = g(x)
[n + 3 is even.]

So, g(x) is an even function.


Correct answer : (1)
 2.  
Write the domain of the function f(x) = 9x.
a.
(- ∞ , 0]
b.
(0, ∞)
c.
( - ∞ , 0)
d.
[0, ∞)


Solution:

Since f(x) = 9x is defined for all x [0,∞), the domain of the function is [0,∞).


Correct answer : (4)
 3.  
Find the constant of variation for the function f (x) = 6(x11).
a.
6
b.
1 11
c.
11


Solution:

f (x) = 6x11 = 6x111

= 6 · (x)111

On comparing f(x) = 6 · (x)111 with standard power function f (x) = k (x)a, we have k = 6, a = 1 / 11

The constant of variation of the given function is k = 6.
[The constant of variation of f (x) = k (x)a is k.]


Correct answer : (1)
 4.  
Find the value of f(32), if f(x) = 28 x- 15.
a.
28
b.
19
c.
42
d.
14


Solution:

f(x) = 28x- 15

f(32) = 28(32)- 15
[Substitute x = 32.]

= 28 (25)- 15
[Use 25 = 32.]

(28) (2)- 1
[Use (am)n = amn.]

= (28)(1 / 2) = 14.
[Use a- 1 = 1a.]


Correct answer : (4)
 5.  
Find the power for the function g(x) = 9x4.
a.
4
b.
- 4
c.
9


Solution:

g(x) = 9x4

= 9 · x - 4

= 9 · x - 4

On comparing g(x) = 9 · x - 4 with the standard power function f (x) = k · (x)a, we have k = 9, a = - 4.

The power of the given function is a = - 4
[The power of the function f (x) = k · (x)a is a.]


Correct answer : (2)
 6.  
Let f(x) = 8x- 17. Which of the following is correct?
a.
f is a constance function
b.
f is neither an even nor an odd function
c.
f is an even function
d.
f is an odd function


Solution:

Clearly, f(x) = 8x- 17 is a power function.

f(- x) = 8(- x)- 17 = 8(- x)17 = - 8x17 = - 8x- 17 = - f(x)

So, f is an odd function.


Correct answer : (4)
 7.  
Let f(x) = 2x12. Which of the following is correct?
a.
f(x) is a constant function
b.
f(x) is an odd function
c.
f(x) is neither an odd function nor an even function
d.
f(x) is an even function


Solution:

Clearly, f(x) = 2x12 is a power function.

f(- x) = 2 (- x)12 = 2x12 = f(x)

So, f(x) is an even function.


Correct answer : (4)
 8.  
Choose the monomial function from the following.
a.
y = 3x- 5
b.
y = 6x15
c.
y = 5x4
d.
y = x- 15


Solution:

Any function that can be written as f(x) = k (or) f (x) = k · xn where k is a constant and n is a positive integer, is a monomial function.
[Definition of monomial function.]

Clearly, from the given choices, y = 5x4 is a monomial function.


Correct answer : (3)
 9.  
Let f(x) = 3x3, which of the following is correct?
a.
f(x) is an function
b.
The graph of f(x) is symmetric with respect to x - axis
c.
The graph of f(x) is symmetric with respect to the origin
d.
The graph of f(x) symmetric with respect to y - axis


Solution:

f(x) = 3x3

f(- x) = 3(- x)3 = - 3x3 = - f(x) and hence f(x) is an odd function.

Since f(x) is an odd function, the graph of it is symmetric with respect to the origin.


Correct answer : (3)
 10.  
Let f(x) = - x0.8. Which of the following is correct?
a.
The graph of f(x) is symmetric with respect to the x - axis
b.
The graph of f(x) is symmetric with respect to the y - axis
c.
f(x) is an odd function
d.
The graph of f(x) is symmetric with respect to the origin


Solution:

f(x) = - x0.8

f(- x) = - x0.8 = - (- x)45

= - (- x 5)4
[Use amn = (an)m.]

= - (- x5)4

= - (x5)4 = - x0.8 = f(x)

So, f(x) is an even function and hence its graph is symmetric with respect to the y - axis.


Correct answer : (2)

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