The function given is:

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

This is an exponential decay function. Let's break down its features.

1. Function Type:

The function is exponential, specifically an exponential decay function, because the base $\frac{1}{3}$ is less than 1, which causes the graph to decrease as $x$ increases.

2. Asymptote:

• The horizontal asymptote occurs where the function tends to as $x \to \infty$.
• From the equation, you can see the graph is shifted down by 3 units, so the horizontal asymptote is at $y = -3$.

3. Range:

The range of the function is all values above the asymptote. Since the function never touches $y = -3$, the range is $(-3, \infty)$.

4. Domain:

• The domain of an exponential function like this one is all real numbers, $(-\infty, \infty)$, because the function is defined for any $x$.

5. End Behavior:

• Left-side behavior: As $x \to -\infty$, the function grows large, moving upward toward $\infty$.
• Right-side behavior: As $x \to \infty$, the function approaches the asymptote $y = -3$ from above but never reaches it.

Now let's fill in the blanks for your answer form:

1. The function $f(x)$ is an exponential function with a horizontal asymptote of -3.
2. The range of the function is $(-3, \infty)$, and it is decreasing on its domain of $(-\infty, \infty)$.
3. The end behavior on the left side is as $x \to -\infty$, $f(x) \to \infty$.
4. The end behavior on the right side is as $x \to \infty$, $f(x) \to -3$.

Would you like further details or have any questions?

Here are 5 related questions to expand on the topic:

1. How does the base of an exponential function affect its growth or decay?
2. What transformations are applied to the parent function $y = 2 \left( \frac{1}{3} \right)^x$?
3. What are the key differences between exponential growth and decay functions?
4. How do vertical shifts impact the range of a function?
5. How can you determine the end behavior of any exponential function?

Tip: Always check the horizontal asymptote and the transformations applied to the basic exponential function to understand the behavior of the graph.