Question

an exponential function \\(f(x)\\) is reflected across the \\(x\\)-axis to create the function \\(g(x)\\). which is a true statement regarding \\(f(x)\\) and \\(g(x)\\)?

• the two functions have the same initial value.
• the two functions will cross each other on the axis.
• the two functions have reciprocal output values of each other for any given input value.
• the two functions have opposite output values of each other for any given input value.

Explanation:

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<reasoning>

Define the relationship between the functions

\[
g(x) = -f(x)
\]

Analyze the output values

\[
\text{For any input } x, \text{ the output of } g(x) \text{ is the negative (opposite) of } f(x).
\]

Evaluate the given statements

\[

$$\begin{aligned} &\text{Statement 1: Same initial value } (f(0) = g(0)) \implies f(0) = -f(0) \implies f(0) = 0 \text{ (False for standard exponential functions } f(x) = ab^x \text{ where } a eq 0). \\ &\text{Statement 2: Cross each other on the axis } \implies f(x) = g(x) = 0 \text{ (False, exponential functions do not cross the x-axis asymptote } y=0). \\ &\text{Statement 3: Reciprocal output values } \implies g(x) = \frac{1}{f(x)} \text{ (False). } \\ &\text{Statement 4: Opposite output values } \implies g(x) = -f(x) \text{ (True). } \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>The two functions have the same initial value.</mcq-option>
<mcq-option>The two functions will cross each other on the axis.</mcq-option>
<mcq-option>The two functions have reciprocal output values of each other for any given input value.</mcq-option>
<mcq-correct>The two functions have opposite output values of each other for any given input value.</mcq-correct>
</answer>

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Answer:

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<reasoning>

Define the relationship between the functions

\[
g(x) = -f(x)
\]

Analyze the output values

\[
\text{For any input } x, \text{ the output of } g(x) \text{ is the negative (opposite) of } f(x).
\]

Evaluate the given statements

\[

$$\begin{aligned} &\text{Statement 1: Same initial value } (f(0) = g(0)) \implies f(0) = -f(0) \implies f(0) = 0 \text{ (False for standard exponential functions } f(x) = ab^x \text{ where } a eq 0). \\ &\text{Statement 2: Cross each other on the axis } \implies f(x) = g(x) = 0 \text{ (False, exponential functions do not cross the x-axis asymptote } y=0). \\ &\text{Statement 3: Reciprocal output values } \implies g(x) = \frac{1}{f(x)} \text{ (False). } \\ &\text{Statement 4: Opposite output values } \implies g(x) = -f(x) \text{ (True). } \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>The two functions have the same initial value.</mcq-option>
<mcq-option>The two functions will cross each other on the axis.</mcq-option>
<mcq-option>The two functions have reciprocal output values of each other for any given input value.</mcq-option>
<mcq-correct>The two functions have opposite output values of each other for any given input value.</mcq-correct>
</answer>

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