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To find the range (σύνολο τιμών) of the function \( f(x) = \frac{2 + \ln x}{x} \), let's analyze it step by step.

### 1. Domain Analysis
First, we identify the domain of the function. The function \( f(x) \) involves the natural logarithm \( \ln x \), which is defined for \( x > 0 \). Thus, the domain of \( f(x) \) is \( (0, \infty) \).

### 2. Derivative of \( f(x) \)
To find the range, it's helpful to determine where the function increases or decreases. We do this by finding the first derivative of \( f(x) \) and analyzing its sign.

Given:
\[
f(x) = \frac{2 + \ln x}{x}
\]

The first derivative \( f'(x) \) is obtained using the quotient rule:
\[
f'(x) = \frac{(x \cdot \frac{1}{x} - (2 + \ln x) \cdot 1)}{x^2}
\]

Simplifying the numerator:
\[
f'(x) = \frac{1 - (2 + \ln x)}{x^2} = \frac{-1 - \ln x}{x^2}
\]

### 3. Critical Points
We set \( f'(x) = 0 \) to find the critical points:
\[
\frac{-1 - \ln x}{x^2} = 0
\]

This equation implies:
\[
-1 - \ln x = 0 \quad \Rightarrow \quad \ln x = -1 \quad \Rightarrow \quad x = \frac{1}{e}
\]

This is a critical point. To determine whether it is a maximum or minimum, we analyze the sign of \( f'(x) \).

### 4. Behavior of \( f'(x) \)
- **For \( 0 < x < \frac{1}{e} \)**: \( \ln x < -1 \), so \( f'(x) > 0 \), meaning \( f(x) \) is increasing.
- **For \( x > \frac{1}{e} \)**: \( \ln x > -1 \), so \( f'(x) < 0 \), meaning \( f(x) \) is decreasing.

Thus, \( f(x) \) has a maximum at \( x = \frac{1}{e} \).

### 5. Maximum Value
Substitute \( x = \frac{1}{e} \) into the function to find the maximum value:
\[
f\left(\frac{1}{e}\right) = \frac{2 + \ln\left(\frac{1}{e}\right)}{\frac{1}{e}} = e \cdot \left(2 + \ln\left(\frac{1}{e}\right)\right)
\]
\[
= e \cdot (2 - 1) = e \cdot 1 = e
\]

### 6. Range of \( f(x) \)
- As \( x \to 0^+ \), \( \ln x \to -\infty \), so \( f(x) \to -\infty \).
- As \( x \to \infty \), \( f(x) \to 0^+ \).

Thus, the range of \( f(x) \) is \( (-\infty, e] \).

Would you like further details on any part of this solution or have any other questions?

### Relative Questions:
1. What are the steps for finding the critical points of a function?
2. How does the behavior of the derivative indicate whether a function is increasing or decreasing?
3. What is the significance of the logarithm function's domain in analyzing \( f(x) \)?
4. Can we apply the same method to determine the range of more complex functions?
5. How do limits help in determining the range of a function?

### Tip:
When dealing with logarithmic functions, always remember to check the domain first, as it directly impacts the overall analysis of the function's behavior.

Learn how to find the range (σύνολο τιμών) of the function f(x) = (2 + ln x) / x step-by-step. Discover its domain, critical points, derivative behavior, and the resulting range. Ideal for advanced high school and college students studying functions and derivatives.

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