Question: Complete the table of values for the functions $f(x) = |x - 3| - 4$ and $g(x) = 3^x - 5$.
Write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.
$x \quad f(x) \quad g(x)$
$0 \quad -1 \quad -4$
$1$
$2 \quad -3 \quad 4$
$3$
$4
Based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?
$x = 1$
between $x = 1$ and $x = 2$
$x = 2$
between $x = 2$ and $x = 3$
1. **State the problem:** We need to complete the table for $f(x) = |x - 3| - 4$ and $g(x) = 3^x - 5$ at $x = 1, 3, 4$ and determine where $f(x) = g(x)$ has a solution.
2. **Calculate $f(x)$ values:**
- For $x=1$: $$f(1) = |1 - 3| - 4 = | -2 | - 4 = 2 - 4 = -2$$
- For $x=3$: $$f(3) = |3 - 3| - 4 = 0 - 4 = -4$$
- For $x=4$: $$f(4) = |4 - 3| - 4 = 1 - 4 = -3$$
3. **Calculate $g(x)$ values:**
- For $x=1$: $$g(1) = 3^1 - 5 = 3 - 5 = -2$$
- For $x=3$: $$g(3) = 3^3 - 5 = 27 - 5 = 22$$
- For $x=4$: $$g(4) = 3^4 - 5 = 81 - 5 = 76$$
4. **Complete the table:**
\begin{tabular}{c|c|c}
$x$ & $f(x)$ & $g(x)$ \\
\hline
0 & -1 & -4 \\
1 & -2 & -2 \\
2 & -3 & 4 \\
3 & -4 & 22 \\
4 & -3 & 76 \\
\end{tabular}
5. **Determine where $f(x) = g(x)$:**
- At $x=1$, $f(1) = -2$ and $g(1) = -2$, so $f(1) = g(1)$.
- Between $x=1$ and $x=2$, $f(x)$ decreases from $-2$ to $-3$, $g(x)$ increases from $-2$ to $4$, so they are not equal in between.
- At $x=2$, $f(2) = -3$ and $g(2) = 4$, not equal.
- Between $x=2$ and $x=3$, $f(x)$ decreases from $-3$ to $-4$, $g(x)$ increases from $4$ to $22$, no equality.
**Final answer:** The equation $f(x) = g(x)$ has a solution at $x = 1$.