Question

the graph of (f(x) = \frac{1}{2}(2.5)^x) and its reflection across the x-axis, (g(x)), are shown.

what is the range of (g(x))?

• all real numbers
• all real numbers less than 0
• all real numbers greater than 0
• all real numbers less than or equal to 0

Explanation:

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<reasoning>

Identify the function and its transformation

\[

$$\begin{aligned} f(x) &= \frac{1}{2}(2.5)^x \\ g(x) &= -f(x) = -\frac{1}{2}(2.5)^x \end{aligned}$$

\]

Determine the range of the transformed function

\[

$$\begin{aligned} &\text{Since } (2.5)^x > 0 \text{ for all real } x: \\ &g(x) < 0 \text{ for all real } x \end{aligned}$$

\]

Match with the given options

\[
\text{Range of } g(x) \text{ is } (-\infty, 0), \text{ which corresponds to "all real numbers less than 0".}
\]
</reasoning>

<answer>
<mcq-option>all real numbers</mcq-option>
<mcq-correct>all real numbers less than 0</mcq-correct>
<mcq-option>all real numbers greater than 0</mcq-option>
<mcq-option>all real numbers less than or equal to 0</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the function and its transformation

\[

$$\begin{aligned} f(x) &= \frac{1}{2}(2.5)^x \\ g(x) &= -f(x) = -\frac{1}{2}(2.5)^x \end{aligned}$$

\]

Determine the range of the transformed function

\[

$$\begin{aligned} &\text{Since } (2.5)^x > 0 \text{ for all real } x: \\ &g(x) < 0 \text{ for all real } x \end{aligned}$$

\]

Match with the given options

\[
\text{Range of } g(x) \text{ is } (-\infty, 0), \text{ which corresponds to "all real numbers less than 0".}
\]
</reasoning>

<answer>
<mcq-option>all real numbers</mcq-option>
<mcq-correct>all real numbers less than 0</mcq-correct>
<mcq-option>all real numbers greater than 0</mcq-option>
<mcq-option>all real numbers less than or equal to 0</mcq-option>
</answer>

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