Graph the exponential function g(x) = 4x - 1. Plot two points on the graph of the function and also draw the asymptote. Additionally give the domain and range using interval notation

To graph the exponential function \( g(x) = 4^x - 1 \), we'll go through the following steps:

### 1. **Understand the function:**
   - The function \( g(x) = 4^x - 1 \) is an exponential function with a base of 4 and a vertical shift downward by 1 unit.

### 2. **Identify key points:**
   - Let's find two points on the graph by plugging in specific values for \( x \).

     - For \( x = 0 \):
       \[
       g(0) = 4^0 - 1 = 1 - 1 = 0
       \]
       So, one point is \( (0, 0) \).

     - For \( x = 1 \):
       \[
       g(1) = 4^1 - 1 = 4 - 1 = 3
       \]
       So, another point is \( (1, 3) \).

### 3. **Asymptote:**
   - The horizontal asymptote for the function is \( y = -1 \). This is because as \( x \) approaches negative infinity, \( 4^x \) approaches 0, making \( g(x) \) approach \( -1 \).

### 4. **Graphing:**
   - Now, you can plot the points \( (0, 0) \) and \( (1, 3) \) on the coordinate plane.
   - Draw the curve of the exponential function that passes through these points.
   - Draw the horizontal asymptote at \( y = -1 \).

### 5. **Domain and Range:**
   - **Domain**: The function \( g(x) = 4^x - 1 \) is defined for all real numbers \( x \), so the domain is \( (-\infty, \infty) \).
   - **Range**: Since the function is always greater than \( -1 \) (because \( 4^x \) is always positive and shifts downward by 1), the range is \( (-1, \infty) \).

Here's a summary of the key points:

- **Points**: \( (0, 0) \) and \( (1, 3) \)
- **Asymptote**: \( y = -1 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-1, \infty) \)

Would you like me to generate a graph of this function for visual reference?

Learn how to graph the exponential function g(x) = 4^x - 1 with detailed steps for plotting points, identifying the asymptote, and determining the domain and range. Suitable for advanced high school students in Grades 10-12.

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