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

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 $(0, 2)$.

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.

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.

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!