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The functions
f(x)=5(2)^(x) and
g(x)=5(b)^(x) are graphed in the
xy-plane. If the graph of function
g is always increasing and
f(x) > g(x) for all
x > 0, then which of the following could be the value of
b ?
Choose 1 answer:
(A) 0.25
(B) 1.25
(c) 2
(D) 5

The functions $f(x)=5(2)^{x}$ and $g(x)=5(b)^{x}$ are graphed in the $x y$ -plane. If the graph of function $g$ is always increasing and f(x)>g(x) for all x>0 , then which of the following could be the value of $b$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ . $25$ $\newline$ (B) $1$ . $25$ $\newline$ (C) $2$ $\newline$ (D) $5$

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Compare linear and exponential growth

Full solution

Q. The functions

f(x)=5(2)^{x}

and

g(x)=5(b)^{x}

are graphed in the

x y

-plane. If the graph of function

g

is always increasing and

f(x)>g(x)

for all

x>0

, then which of the following could be the value of

b

?

\newline

Choose

1

answer:

\newline

(A)

0

.

25

\newline

(B)

1

.

25

\newline

(C)

2

\newline

(D)

5

Analyze Functions: Analyze the given functions and conditions. $\newline$ We have two functions $f(x) = 5(2)^x$ and $g(x) = 5(b)^x$ . We are told that f(x) > g(x) for all x > 0, which means that for any positive $x$ , the value of $f(x)$ is greater than the value of $g(x)$ . Additionally, the graph of $g$ is always increasing, which implies that $b$ must be greater than $1$ because if $b$ were less than or equal to $1$ , the graph of $g$ would not be increasing.
Compare Bases: Compare the bases of the exponential functions. $\newline$ Since $f(x) = 5(2)^x$ and $g(x) = 5(b)^x$ , and we know that f(x) > g(x) for all x > 0, the base of $f(x)$ , which is $2$ , must be greater than the base of $g(x)$ , which is $b$ . This means that $b$ must be less than $2$ .
Combine Conditions: Combine the conditions to find the possible range for $b$ . From Step $1$ , we know that b > 1, and from Step $2$ , we know that b < 2. Therefore, the value of $b$ must be between $1$ and $2$ .
Evaluate Choices: Evaluate the answer choices. $\newline$ We are given four options for the value of $b$ : $\newline$ (A) $0.25$ $\newline$ (B) $1.25$ $\newline$ (C) $2$ $\newline$ (D) $5$ $\newline$ From our previous steps, we know that $b$ must be greater than $1$ and less than $2$ . Therefore, the only option that fits this condition is (B) $1.25$ .

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\newline

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Both of these functions grow as

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f(x)

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\newline

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\newline

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\newline

Simplify any fractions.

\newline

h'(8)=

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Question

p(x)

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\newline

Simplify any fractions.

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u(x)

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a(x)

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\newline

c(x)

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\newline

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c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

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Question

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\newline

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\newline

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o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

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