https://file.aiquickdraw.com/mathbot/file/1726317781892b74fvmb3.jpg

You are asked to graph the exponential function \( g(x) = 2^{x-3} \), plot two points on the graph, identify the asymptote, and determine the domain and range using interval notation.

### Steps to solve:

1. **Equation of the function**:  
   The function is \( g(x) = 2^{x-3} \). This is an exponential function where the base is 2, and the exponent is \( x - 3 \). The graph is a transformation of the basic exponential function \( 2^x \).

2. **Plotting two points**:  
   To find two points, we can substitute values of \( x \) into the function.

   - For \( x = 3 \):
     \[
     g(3) = 2^{3-3} = 2^0 = 1
     \]
     So, one point is \( (3, 1) \).

   - For \( x = 4 \):
     \[
     g(4) = 2^{4-3} = 2^1 = 2
     \]
     Another point is \( (4, 2) \).

   These two points are \( (3, 1) \) and \( (4, 2) \).

3. **Asymptote**:  
   The asymptote of an exponential function \( a^{x-h} \) is horizontal. Since there is no vertical shift in the given function, the asymptote will be at:
   \[
   y = 0
   \]
   This is because, as \( x \to -\infty \), the function \( g(x) = 2^{x-3} \to 0 \).

4. **Domain**:  
   The domain of an exponential function is always all real numbers, as there are no restrictions on \( x \):
   \[
   \text{Domain: } (-\infty, \infty)
   \]

5. **Range**:  
   The range of this function is all positive values since an exponential function like \( 2^{x-3} \) never touches or crosses the \( x \)-axis but approaches 0:
   \[
   \text{Range: } (0, \infty)
   \]

Would you like me to create a graph for this function or go into further details? 

Here are 5 related questions for further exploration:
1. What transformations occur to shift the function \( g(x) = 2^x \) to \( g(x) = 2^{x-3} \)?
2. How does changing the base of the exponential function affect its graph?
3. How can you determine the asymptote of an exponential function with a vertical shift?
4. What are the key differences between exponential growth and exponential decay functions?
5. How would the graph of \( g(x) = 2^{x-3} \) change if a negative coefficient was added, e.g., \( g(x) = -2^{x-3} \)?

**Tip:** When graphing exponential functions, always start by identifying key points like the y-intercept and transformations for easy graphing.

Graph the exponential function g(x) = 2^(x-3). To do this, plot two points on the graph of the function, and also draw the asymptote. Additionally, give the domain and range of the function using interval notation.

Learn how to graph the exponential function g(x) = 2^(x-3) by plotting key points, identifying the horizontal asymptote, and determining its domain and range using interval notation. Ideal for high school students, this problem explains key concepts in exponential functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation