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:

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.