We are tasked with finding the domain and range of the exponential function:

$f(x) = 2 \left(\frac{1}{3}\right)^x$

Step 1: Domain of an Exponential Function

The domain of any exponential function, such as $f(x) = a \cdot b^x$, is all real numbers. This is because $x$ can take any value, whether positive, negative, or zero.

Thus, the domain is: $\boxed{\text{All real numbers} \, (-\infty, \infty)}$

Step 2: Range of the Function

The base of the exponential function is $\frac{1}{3}$, which is positive. Exponential functions with positive bases never result in negative outputs. Additionally:

• $f(x)$ approaches $0$ as $x \to \infty$ (positive direction), because the term $\left(\frac{1}{3}\right)^x$ gets smaller and smaller but never reaches $0$.
• $f(x) \to \infty$ as $x \to -\infty$ (negative direction), because $\left(\frac{1}{3}\right)^x$ grows as the exponent becomes more negative (it behaves like $3^{-x}$, which increases as $x \to -\infty$).

The range is all positive real numbers: $\boxed{\text{All positive real numbers} \, (0, \infty)}$

Final Answer:

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$

Would you like to see a graph or further explanation?

Related Questions:

1. How does the base $b$ affect the behavior of exponential functions?
2. What happens to the range if we multiply $f(x)$ by a negative constant?
3. How does the graph of $f(x)$ behave as $x \to \infty$ and $x \to -\infty$?
4. How would the range change if the function included a vertical shift?
5. How can exponential functions model real-world problems, like population growth or radioactive decay?

Tip: Always check whether the base of the exponential function is greater than or less than 1. It determines whether the function grows or decays!