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The function given is:

\[
f(x) = 2 + \log_3{x}
\]

This is a logarithmic function with a vertical shift. Let's analyze it step-by-step.

### 1. Understanding the Function
The base of the logarithm is 3, so the function \( \log_3{x} \) has a basic shape of a logarithmic curve, which generally starts from \( x = 0 \) (approaching a vertical asymptote there) and increases slowly as \( x \) grows. The "+ 2" means the entire graph of \( \log_3{x} \) is shifted up by 2 units.

### 2. Vertical Asymptote
For the function \( f(x) = 2 + \log_3{x} \), the vertical asymptote is at \( x = 0 \), since logarithmic functions are undefined for \( x \leq 0 \). This asymptote will be a vertical line at \( x = 0 \).

### 3. Plotting Points
Let's find two points on the graph by substituting values for \( x \):

- **For \( x = 1 \):**
  \[
  f(1) = 2 + \log_3{1} = 2 + 0 = 2
  \]
  So, one point is \( (1, 2) \).

- **For \( x = 3 \):**
  \[
  f(3) = 2 + \log_3{3} = 2 + 1 = 3
  \]
  So, another point is \( (3, 3) \).

### Summary of Key Information
- **Vertical Asymptote**: \( x = 0 \)
- **Points on the Graph**: \( (1, 2) \) and \( (3, 3) \)

Would you like a graph of this function or more details on any part of the solution?

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### Related Questions
1. What is the effect of changing the base of the logarithm on the graph?
2. How does the function \( f(x) = 2 + \log_b{x} \) change if we add a coefficient to \( \log_b{x} \)?
3. What is the domain and range of \( f(x) = 2 + \log_3{x} \)?
4. How would we find the x-intercept of \( f(x) = 2 + \log_3{x} \)?
5. What happens to the graph if we replace \( \log_3{x} \) with \( \log_3{(x - h)} \) for some constant \( h \)?

### Tip
For logarithmic functions of the form \( f(x) = \log_b{x} + c \), the constant \( c \) shifts the graph vertically, and the base \( b \) affects the rate of growth.

Graph the logarithmic function \( f(x) = 2 + \log_3{x} \). Plot two points on the graph of the function, and also draw the asymptote.

Learn how to graph the logarithmic function f(x) = 2 + log_3(x) by plotting points and identifying the vertical asymptote. This problem covers concepts like transformations and shifts in logarithmic functions. Suitable for high school students in Grades 10-12.

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