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Functions
f(x)=-2((1)/(3))^(x) and
g(x)=2((1)/(3))^(x) are graphed in the
xy-plane. If
(a,b) is a point on the graph of
f, then which of the following is a point on the graph of
g ?
Choose 1 answer:
(A)
(a,b)
(B)
(-a,b)
(c)
(a,-b)
(D)
(-a,-b)

Functions $f(x)=-2\left(\frac{1}{3}\right)^{x}$ and $g(x)=2\left(\frac{1}{3}\right)^{x}$ are graphed in the $x y$ -plane. If $(a, b)$ is a point on the graph of $f$ , then which of the following is a point on the graph of $g$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $(a, b)$ $\newline$ (B) $(-a, b)$ $\newline$ (C) $(a,-b)$ $\newline$ (D) $(-a,-b)$

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Compare linear and exponential growth

Full solution

Q. Functions

f(x)=-2\left(\frac{1}{3}\right)^{x}

and

g(x)=2\left(\frac{1}{3}\right)^{x}

are graphed in the

x y

-plane. If

(a, b)

is a point on the graph of

f

, then which of the following is a point on the graph of

g

?

\newline

Choose

1

answer:

\newline

(A)

(a, b)

\newline

(B)

(-a, b)

\newline

(C)

(a,-b)

\newline

(D)

(-a,-b)

Functions and coefficients: We have two functions: $\newline$ $f(x) = -2 \cdot \left(\frac{1}{3}\right)^x$ $\newline$ $g(x) = 2 \cdot \left(\frac{1}{3}\right)^x$ $\newline$ We need to determine the relationship between a point $(a, b)$ on the graph of $f$ and the corresponding point on the graph of $g$ .
Definition of $f(a)$ : Since $f(x)$ and $g(x)$ are both functions of $(\frac{1}{3})^x$ , they have the same base for their exponential terms. The difference between $f(x)$ and $g(x)$ is the coefficient in front of the exponential term. For $f(x)$ , the coefficient is $-2$ , and for $g(x)$ , it is $2$ .
Finding $g(a)$ : If $(a, b)$ is a point on the graph of $f$ , then by definition, $f(a) = b$ . This means that $b = -2 \cdot \left(\frac{1}{3}\right)^a$ .
Comparison of $f(a)$ and $g(a)$ : To find the corresponding point on the graph of $g$ , we need to find $g(a)$ . Since $g(x) = 2 \cdot \left(\frac{1}{3}\right)^x$ , we have $g(a) = 2 \cdot \left(\frac{1}{3}\right)^a$ .
Corresponding point on g graph: Comparing $f(a) = -2 \times \left(\frac{1}{3}\right)^a$ and $g(a) = 2 \times \left(\frac{1}{3}\right)^a$ , we can see that $g(a)$ is the negative of $f(a)$ . Therefore, if $f(a) = b$ , then $g(a) = -b$ .
Corresponding point on g graph: Comparing $f(a) = -2 \times \left(\frac{1}{3}\right)^a$ and $g(a) = 2 \times \left(\frac{1}{3}\right)^a$ , we can see that $g(a)$ is the negative of $f(a)$ . Therefore, if $f(a) = b$ , then $g(a) = -b$ .The corresponding point on the graph of $g$ for the point $(a, b)$ on the graph of $f$ is $(a, -b)$ . This means that the correct answer is (C) $(a, -b)$ .

More problems from Compare linear and exponential growth

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Jill bought a used motorcycle from a seller online for $1,200. The seller will charge her $5 a day to store the motorcycle at his house until she is able to pick it up. You can use a function to describe the total amount of money Jill will owe the seller if she waits

x

days to pick up the motorcycle.

\newline

Write an equation for the function. If it is linear, write it in the form

g(x) = mx + b

. If it is exponential, write it in the form

g(x) = a(b)^x

\newline

g(x) = \underline{\hspace{3em}}

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f(t)=-2t-7

change over the interval from

t=-7

t=-6

\newline

\newline

\left[\text{f(t) decreases by } 2\right]

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\left[\text{f(t) increases by } 2\right]

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g(t)=9^t

change over the interval from

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\left[\text{g(t) decreases by a factor of } 81\right]

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\left[\text{g(t) increases by a factor of } 81\right]

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\left[\text{g(t) increases by } 18\%\right]

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\left[\text{g(t) decreases by } 9\%\right]

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Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A)f(x) = 8x + 3.3

\newline

(B)g(x) = 3.3^x - 5

Posted 9 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A) \ f(x) = 2x + 9

\newline

(B)\ g(x) = 2^x

Posted 9 months ago

Question

f(x)

g(x)

are differentiable.

\newline

h(x)

h(x)=\frac{g(x)}{f(x)}

\newline

f(8)=1

f'(8)=-6

g(8)=8

g'(8)=-10

h'(8)

\newline

Simplify any fractions.

\newline

h'(8)=

Posted 8 months ago

Question

p(x)

q(x)

are differentiable.

\newline

r(x)

r(x)= \frac{p(x)}{q(x)}

\newline

p(3)= 2

p'(3)= 4

q(3)= 6

q'(3)= 9

r'(3)

\newline

Simplify any fractions.

\newline

r'(3)=

Posted 9 months ago

Question

u(x)

v(x)

are differentiable.

\newline

w(x)

w(x)= \frac{u(x)}{v(x)}

\newline

u(5)= 3

u'(5)= -2

v(5)= 7

v'(5)= 1

w'(5)

\newline

Simplify any fractions.

\newline

w'(5)=

Posted 9 months ago

Question

a(x)

b(x)

are differentiable.

\newline

c(x)

c(x)= \frac{a(x)}{b(x)}

\newline

a(2)= 4

a'(2)= 5

b(2)= 1

b'(2)= -3

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

Posted 9 months ago

Question

m(x)

n(x)

are differentiable.

\newline

o(x)

o(x)= \frac{m(x)}{n(x)}

\newline

m(7)= 2

m'(7)= -1

n(7)= 4

n'(7)= 8

o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

Posted 9 months ago

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