The given function is $g(x) = 3^x - 3$, which is an exponential function. Here's how we can approach this problem step by step:

1. Graphing the Function

We need to plot the graph of $g(x) = 3^x - 3$. Let's first calculate a few points on the graph:

• At $x = 0$: $g(0) = 3^0 - 3 = 1 - 3 = -2$ So, one point is $(0, -2)$.

• At $x = 1$: $g(1) = 3^1 - 3 = 3 - 3 = 0$ Another point is $(1, 0)$.

• At $x = -1$: $g(-1) = 3^{-1} - 3 = \frac{1}{3} - 3 = \frac{1}{3} - \frac{9}{3} = -\frac{8}{3}$ This gives the point $(-1, -\frac{8}{3})$, or approximately $(-1, -2.67)$.

2. Asymptote

The horizontal asymptote of this function is $y = -3$, because as $x \to -\infty$, $3^x$ approaches 0, so $g(x)$ approaches $-3$.

3. Domain and Range

• Domain: The domain of an exponential function is all real numbers, so the domain is $(-\infty, \infty)$.

• Range: Since the graph never reaches $-3$ but approaches it as $x \to -\infty$, and goes to infinity as $x \to \infty$, the range is $(-3, \infty)$.

Summary of Answers:

• Domain: $(-\infty, \infty)$
• Range: $(-3, \infty)$
• Horizontal Asymptote: $y = -3$
• Points to plot: $(0, -2)$, $(1, 0)$, $(-1, -2.67)$

Would you like further details on this or have any questions?

Here are 5 related questions to expand on this information:

1. How does changing the base of the exponential function affect its graph?
2. What happens to the graph of $g(x) = 3^x - 3$ if we add a constant to $x$, like in $g(x) = 3^{x+1} - 3$?
3. Can you find the y-intercept of any exponential function in general form?
4. What is the effect of a vertical shift on the range of an exponential function?
5. How can you determine the asymptote of a transformed exponential function?

Tip: When graphing exponential functions, always look for the horizontal asymptote first to understand how the function behaves at extreme values of $x$.