# Power Functions with Modeling Worksheet

Power Functions with Modeling Worksheet
• Page 1
1.
If $f$($x$) = $x$$n$ , $n$ is a positive integer is an odd function, then which of the following is correct for the function $g$($x$) = $x$$n$+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.

2.
Write the domain of the function $f$($x$) = 9$\sqrt{x}$.
 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,∞).

3.
Find the constant of variation for the function $f$ ($x$) = 6($\sqrt[11]{x}$).
 a. 6 b. $\frac{1}{11}$ 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.]

4.
Find the value of $f$(32), if $f$($x$) = 28 .
 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.]

5.
Find the power for the function $g$($x$) = $\frac{9}{{x}^{4}}$.
 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.]

6.
Let $f$($x$) = 8$x$- 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.

7.
Let $f$($x$) = 2$x$12. 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.

8.
Choose the monomial function from the following.
 a. $y$ = 3$x$- 5 b. $y$ = 6${x}^{\frac{1}{5}}$ c. $y$ = 5$x$4 d. $y$ =

#### 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.

9.
Let $f$($x$) = 3$x$3, 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.

10.
Let $f$($x$) = - $x$0.8. 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.