https://file.aiquickdraw.com/mathbot/file/1726323355141gkenely7.jpg

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 \( x \to -\infty \).

Graph the exponential function g(x) = 2^x - 1. To do this, plot two points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button. Additionally, give the domain and range of the function using interval notation.

Learn how to graph the exponential function g(x) = 2^x - 1, including plotting key points and drawing the asymptote. Understand the domain and range of the function in interval notation. This problem covers core exponential function concepts and is suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation