Let’s analyze and solve the question based on the image provided.

1. Function Analysis: The base function is $f(x) = \log_3{x}$, and the transformation is described as $h(x) = 7 + \log_3{x}$.

2. Transformation Explanation:

   • Adding $7$ to $\log_3{x}$ represents a vertical shift upward by 7 units.
   • Hence, the correct answer is B, as indicated.

3. Vertical Asymptote:

   • For $f(x) = \log_3{x}$, the vertical asymptote is at $x = 0$.
   • Since no horizontal transformations (e.g., shifts to the left or right) are applied, the asymptote of $h(x)$ remains $x = 0$.

4. Domain and Range:

   • Domain: The domain of $h(x)$ is $x > 0$ (since the logarithmic function is undefined for $x \leq 0$).
   • Range: Since the graph is shifted up by 7, the range is $(-\infty, \infty)$, as logarithmic functions cover all real values.

Would you like me to clarify anything or assist further? Here are five related questions and a tip:

Related Questions:

1. How does a horizontal shift affect the vertical asymptote of a logarithmic function?
2. What happens to the range of $\log_3{x}$ if multiplied by a constant?
3. How does the base of the logarithm (e.g., 2, 3, 10) impact the shape of the graph?
4. Can you determine the inverse of $h(x) = 7 + \log_3{x}$?
5. How would the graph change if $h(x) = \log_3{(x - 2)} + 7$?

Tip:

When working with logarithmic transformations, always isolate the base graph first and analyze shifts step by step (vertical, horizontal, or scaling). This ensures clarity in determining asymptotes, domains, and ranges.