The given problem asks to graph the exponential function $g(x) = 3^x - 3$, plot points, draw the asymptote, and determine the domain and range of the function.

1. Graphing the Function

The function $g(x) = 3^x - 3$ is an exponential function where:

• The base is $3$, meaning the function grows exponentially.
• The $-3$ indicates a vertical shift downward by 3 units.

Key Points

• The graph has an asymptote at $y = -3$. The graph approaches this value but never touches it.
• For $x = 0$, $g(0) = 3^0 - 3 = 1 - 3 = -2$. So, one point is $(0, -2)$.
• For $x = 1$, $g(1) = 3^1 - 3 = 3 - 3 = 0$. Another point is $(1, 0)$.

Asymptote:

The horizontal asymptote is $y = -3$.

2. Domain and Range

• Domain: Since exponential functions are defined for all real values of $x$, the domain is $(-\infty, \infty)$.
• Range: The function approaches but never reaches $y = -3$, and as $x \to \infty$, the function increases without bound. Hence, the range is $(-3, \infty)$.

Summary:

• Domain: $(-\infty, \infty)$
• Range: $(-3, \infty)$

Would you like a more detailed explanation or help with graphing the function?

Here are 5 related questions to deepen your understanding:

1. How do transformations affect the graph of an exponential function?
2. What happens to the graph of $g(x) = 3^x$ when we add or subtract constants?
3. How do you determine the asymptote of an exponential function?
4. What is the effect of different bases on the growth rate of exponential functions?
5. How does the graph of $3^x$ compare to other exponential functions like $2^x$ or $e^x$?

Tip: The domain of exponential functions is always all real numbers, but the range depends on any vertical shifts!