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:

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.

Answer:

• 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. (Correct answer)