Question: Look at the table of values for the functions $f(x) = 2.5x + 2$ and $g(x) = 4^{x - 2}$.\nx\nf(x)\ng(x)\n0\n2\n\frac{1}{16}\n1\n4.5\n\frac{1}{4}\n2\n7\n1\n3\n9.5\n4\n4\n12\n16\n\nBased on the values in the table, where does the equation $f(x) = g(x)$ have a solution?\n\nbetween $x = 2$ and $x = 3$\n\nx = 3\n\nbetween $x = 3$ and $x = 4$\n\nx = 4
1. **State the problem:** We want to find where the equation $f(x) = g(x)$ has a solution based on the given table values for $f(x) = 2.5x + 2$ and $g(x) = 4^{x - 2}$.\n\n2. **Recall the functions:**\n$$f(x) = 2.5x + 2$$\n$$g(x) = 4^{x - 2}$$\n\n3. **Check the table values:**\n\n| $x$ | $f(x)$ | $g(x)$ |\n|-----|--------|--------|\n| 2 | 7 | 1 |\n| 3 | 9.5 | 4 |\n| 4 | 12 | 16 |\n\n4. **Compare $f(x)$ and $g(x)$ at these points:**\n- At $x=2$, $f(2) = 7$ and $g(2) = 1$, so $f(2) > g(2)$.\n- At $x=3$, $f(3) = 9.5$ and $g(3) = 4$, so $f(3) > g(3)$.\n- At $x=4$, $f(4) = 12$ and $g(4) = 16$, so $f(4) < g(4)$.\n\n5. **Analyze where $f(x)$ and $g(x)$ cross:**\nSince $f(x) > g(x)$ at $x=3$ but $f(x) < g(x)$ at $x=4$, the functions must cross between $x=3$ and $x=4$.\n\n6. **Conclusion:** The equation $f(x) = g(x)$ has a solution between $x = 3$ and $x = 4$.