Let's analyze the problem step by step. The given function is $g(x) = 2^x - 1$.

1. Understanding the Function:

The function $g(x) = 2^x - 1$ is based on the exponential function $f(x) = 2^x$, but it is shifted downward by 1 unit because of the subtraction.

• Base Function: $f(x) = 2^x$ is an exponential growth function.
• Transformation: Subtracting 1 shifts the entire graph downward by 1 unit.

2. Graph Selection:

The graph of $g(x) = 2^x - 1$ starts just below $y = 0$ (approaching $-1$) as $x \to -\infty$ and grows rapidly as $x \to \infty$.

• The correct graph should have:
  • A horizontal asymptote at $y = -1$.
  • Exponential growth for $x > 0$.

By analyzing the graphs provided:

• The first graph (top-left) matches $g(x) = 2^x - 1$.

3. Domain and Range:

• Domain: The exponential function $2^x$ is defined for all real $x$, and shifting it downward does not change its domain.
  • Domain: $(-\infty, \infty)$.
• Range: The range of $f(x) = 2^x$ is $(0, \infty)$. After subtracting 1, the range becomes $(-1, \infty)$.
  • Range: $(-1, \infty)$.

4. Asymptote:

The horizontal asymptote occurs where $y$ approaches a constant value as $x \to -\infty$. For $g(x) = 2^x - 1$, as $x \to -\infty$, $2^x \to 0$, so $g(x) \to -1$.

• Asymptote: $y = -1$.

Final Answer:

• Correct graph: Top-left graph.
• Domain: $(-\infty, \infty)$.
• Range: $(-1, \infty)$.
• Asymptote: $y = -1$.

Do you want further clarification or a deeper explanation of any part? Here are 5 questions to expand on the topic:

1. How does the base of the exponential function (e.g., $2^x$) affect the growth rate?
2. What are the key properties of exponential functions?
3. How do vertical and horizontal shifts impact the graph of a function?
4. What is the difference between exponential growth and decay?
5. How would $g(x) = 2^x + c$ (with different constants $c$) change the graph?

Tip: Always pay close attention to transformations like shifts, reflections, and stretches when analyzing functions!