The task in the image involves graphing the exponential function $g(x) = 2^x - 1$, and also determining its domain, range, and asymptote. Let's break down how to approach this:

1. Graphing the Exponential Function:

The function $g(x) = 2^x - 1$ is an exponential function where the base is 2. Here are the key steps to graph it:

• Key points to plot:

  • At $x = 0$, $g(0) = 2^0 - 1 = 0$.
  • At $x = 1$, $g(1) = 2^1 - 1 = 1$.
  • At $x = -1$, $g(-1) = 2^{-1} - 1 = \frac{1}{2} - 1 = -0.5$.

  Plotting these points will give you a sense of how the curve behaves. You can then sketch the general shape of the graph by extending it on both sides.

• Asymptote:

  • The horizontal asymptote of the function is $y = -1$ because as $x \to -\infty$, the function $2^x$ approaches zero, making $g(x) = 2^x - 1 \to -1$.

2. Domain and Range:

• Domain: The domain of the exponential function is all real numbers, i.e., $(-\infty, \infty)$, since exponential functions are defined for all values of $x$.
• Range: The range of this function is $(-1, \infty)$. The function never actually reaches $y = -1$, but it approaches it, and the function grows infinitely in the positive direction.

Recap of Key Information:

• Graph: Plot the points $(0,0), (1,1), (-1,-0.5)$, and draw a smooth curve through them.
• Asymptote: $y = -1$.
• Domain: $(-\infty, \infty)$.
• Range: $(-1, \infty)$.

Would you like more details or have any questions about this problem?

Here are 5 related questions to explore further:

1. How would the graph change if the function was $g(x) = 2^{x+1} - 1$?
2. How do you find the asymptote of other types of exponential functions?
3. What is the domain and range of $f(x) = 2^{-x}$?
4. How would the function change if the base were a fraction, like $\frac{1}{2}$?
5. How do you graph a transformed function such as $g(x) = 3(2^x - 1)$?

Tip: Exponential functions grow quickly as $x \to \infty$ but approach their asymptotes gradually as $x \to -\infty$.