Question

\\f(x) = 7^x\\
\\g(x) = 7^x + 6\\

which statement about \\(f(x)\\) and its translation, \\(g(x)\\), is true?

• the domain of \\(g(x)\\) is \\(\\{x \mid x > 6\\}\\), and the domain of \\(f(x)\\) is \\(\\{x \mid x > 0\\}\\).
• the domain of \\(g(x)\\) is \\(\\{y \mid y > 0\\}\\), and the domain of \\(f(x)\\) is \\(\\{y \mid y > 6\\}\\).
• the asymptote of \\(g(x)\\) is the asymptote of \\(f(x)\\) shifted six units down.
• the asymptote of \\(g(x)\\) is the asymptote of \\(f(x)\\) shifted six units up.

Explanation:

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<reasoning>

Analyze the parent function

Using the Exponential Functions knowledge point
\[
f(x) = 7^x
\]
The domain is all real numbers, and the horizontal asymptote is \(y = 0\).

Analyze the transformation

Using the Function Transformations knowledge point
\[
g(x) = f(x) + 6 = 7^x + 6
\]
This represents a vertical shift of the entire graph upward by 6 units.

Determine the new asymptote

Using the Horizontal Asymptote knowledge point
\[
\text{Asymptote of } g(x): y = 6
\]
The asymptote of \(g(x)\) is the asymptote of \(f(x)\) shifted six units up.
</reasoning>

<answer>
<mcq-option>The domain of \(g(x)\) is \(\{x \mid x > 6\}\), and the domain of \(f(x)\) is \(\{x \mid x > 0\}\).</mcq-option>
<mcq-option>The domain of \(g(x)\) is \(\{y \mid y > 0\}\), and the domain of \(f(x)\) is \(\{y \mid y > 6\}\).</mcq-option>
<mcq-option>The asymptote of \(g(x)\) is the asymptote of \(f(x)\) shifted six units down.</mcq-option>
<mcq-correct>The asymptote of \(g(x)\) is the asymptote of \(f(x)\) shifted six units up.</mcq-correct>
</answer>

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"Algebra",
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Answer:

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<reasoning>

Analyze the parent function

Using the Exponential Functions knowledge point
\[
f(x) = 7^x
\]
The domain is all real numbers, and the horizontal asymptote is \(y = 0\).

Analyze the transformation

Using the Function Transformations knowledge point
\[
g(x) = f(x) + 6 = 7^x + 6
\]
This represents a vertical shift of the entire graph upward by 6 units.

Determine the new asymptote

Using the Horizontal Asymptote knowledge point
\[
\text{Asymptote of } g(x): y = 6
\]
The asymptote of \(g(x)\) is the asymptote of \(f(x)\) shifted six units up.
</reasoning>

<answer>
<mcq-option>The domain of \(g(x)\) is \(\{x \mid x > 6\}\), and the domain of \(f(x)\) is \(\{x \mid x > 0\}\).</mcq-option>
<mcq-option>The domain of \(g(x)\) is \(\{y \mid y > 0\}\), and the domain of \(f(x)\) is \(\{y \mid y > 6\}\).</mcq-option>
<mcq-option>The asymptote of \(g(x)\) is the asymptote of \(f(x)\) shifted six units down.</mcq-option>
<mcq-correct>The asymptote of \(g(x)\) is the asymptote of \(f(x)\) shifted six units up.</mcq-correct>
</answer>

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