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Let's analyze and solve **Question 6** from the uploaded image.

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The function given is:

\[
f(x) = 2 \cdot \left(\frac{1}{3}\right)^x
\]

### Step 1: Key characteristics of the function
1. **Domain**:
   - The base \( \frac{1}{3} > 0 \), so the exponential function \( \left(\frac{1}{3}\right)^x \) is well-defined for all \(x \in \mathbb{R}\).
   - **Conclusion**: The domain of \(f(x)\) is all real numbers.

2. **Range**:
   - Since \( \left(\frac{1}{3}\right)^x > 0 \) for all \(x\), the product \(2 \cdot \left(\frac{1}{3}\right)^x\) will also always be positive (\(f(x) > 0\)).
   - **Conclusion**: The range is \( f(x) > 0 \), or all positive real numbers.

3. **Asymptotic Behavior**:
   - As \(x \to \infty\), \( \left(\frac{1}{3}\right)^x \to 0 \), and thus \(f(x) \to 0\).
   - As \(x \to -\infty\), \( \left(\frac{1}{3}\right)^x \to \infty\), and thus \(f(x) \to \infty\).
   - **Horizontal Asymptote**: \(y = 0\).

4. **Special Values**:
   - When \(x = 0\), \(f(0) = 2 \cdot 1 = 2\).
   - The graph will pass through \((0, 2)\).

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### Step 2: Analyzing the choices
1. **"The domain of the function is all real numbers."**
   - Correct. The exponential function is defined for all real \(x\).

2. **"The range of the function is all real numbers greater than 0."**
   - Correct. The output of \(f(x)\) is strictly positive.

3. **"As \(x\) increases, the graph of the function approaches the line \(y = \frac{1}{3}\)."**
   - Incorrect. The graph approaches the horizontal asymptote \(y = 0\), not \(y = \frac{1}{3}\).

4. **"The range of the function is all real numbers greater than 2."**
   - Incorrect. The range is all positive real numbers (\(f(x) > 0\)), not limited to values greater than 2.

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### Final Answer:
- The correct options are:
  - **The domain of the function is all real numbers.**
  - **The range of the function is all real numbers greater than 0.**

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Would you like further clarification on any step or additional explanation?  
Here are 5 related questions for practice:  
1. How do you find the horizontal asymptote of an exponential function?  
2. What changes to the range if \(f(x) = 2 \cdot \left(\frac{1}{3}\right)^x - 1\)?  
3. How does the base \(b\) in an exponential function \(b^x\) affect the graph?  
4. Why can’t the function \(f(x) = 2 \cdot \left(\frac{1}{3}\right)^x\) output negative values?  
5. What happens to the graph if \(x\) is replaced by \(-x\) in \(f(x)\)?  

**Tip**: Always check the base of the exponential function to determine the growth or decay behavior!

Consider the graph of the following function: f(x) = 2 * (1/3)^x. Which of the following correctly describes this function? Select all that apply.

Understand the domain, range, and asymptotic behavior of the exponential function f(x) = 2 * (1/3)^x. This problem covers key concepts such as domain as all real numbers, range as positive values, and the horizontal asymptote y = 0. Suitable for Grades 9-11 students learning exponential functions.

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