https://file.aiquickdraw.com/mathbot/file/17264156349009ly8mb2k.jpg

Let's solve this step-by-step:

### 1. **Understand the Functions**
   - \( f(x) = 3x + 1 \)
   - \( g(x) = \left(\frac{2}{3}\right)^x \)

### 2. **Complete the Table**
   Let's compute \( f(x) \) and \( g(x) \) for the given values of \( x \):

   **For \( x = 0 \):**
   - \( f(0) = 3(0) + 1 = 1 \)
   - \( g(0) = \left(\frac{2}{3}\right)^0 = 1 \)

   **For \( x = 2 \):**
   - \( f(2) = 3(2) + 1 = 7 \)
   - \( g(2) = \left(\frac{2}{3}\right)^2 = \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) = \frac{4}{9} \)

   **For \( x = 3 \):**
   - \( f(3) = 3(3) + 1 = 10 \)
   - \( g(3) = \left(\frac{2}{3}\right)^3 = \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) = \frac{8}{27} \)

### 3. **Fill in the Table**
   Based on the calculations:
   
   | \( x \) | \( f(x) \) | \( g(x) \)   |
   |---------|------------|--------------|
   | -1      | -2         | \( \frac{3}{2} \) |
   | 0       | 1          | 1            |
   | 1       | 4          | \( \frac{2}{3} \) |
   | 2       | 7          | \( \frac{4}{9} \) |
   | 3       | 10         | \( \frac{8}{27} \) |

### 4. **Determine Where \( f(x) = g(x) \)**
   We need to find where the equation \( f(x) = g(x) \) has a solution based on the table:
   
   - For \( x = -1 \), \( f(x) = -2 \) and \( g(x) = \frac{3}{2} \). \( f(x) < g(x) \).
   - For \( x = 0 \), \( f(x) = 1 \) and \( g(x) = 1 \). \( f(x) = g(x) \).
   - For \( x = 1 \), \( f(x) = 4 \) and \( g(x) = \frac{2}{3} \). \( f(x) > g(x) \).
   
   The values of \( f(x) \) and \( g(x) \) are equal at \( x = 0 \). Hence, the equation \( f(x) = g(x) \) has a solution at \( x = 0 \).

### 5. **Answer the Question**
   Based on the values in the table, the correct answer is:
   - **\( x = 0 \)**

Would you like further details or have any questions?

### Follow-Up Questions
1. What happens to the functions \( f(x) \) and \( g(x) \) as \( x \) increases?
2. Can you find an interval where \( f(x) < g(x) \)?
3. What kind of function is \( g(x) = \left(\frac{2}{3}\right)^x \)?
4. Why does \( g(x) \) decrease as \( x \) increases?
5. Can there be more than one solution to \( f(x) = g(x) \)?

### Tip
When comparing two functions, plotting them on a graph can provide a clearer visual understanding of where they intersect.

Complete the table of values for the functions f(x) = 3x + 1 and g(x) = (2/3)^x. Based on the values in the table, where does the equation f(x) = g(x) have a solution?

Learn how to solve equations involving linear and exponential functions by completing a table of values for f(x) = 3x + 1 and g(x) = (2/3)^x. This problem explores where these two functions intersect and find the solution using a step-by-step approach. Suitable for students in Grades 8-10.

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