Question: Complete the table of values for the functions $f(x) = \frac{1}{2}|x|$ and $g(x) = 2^x$.
Write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.
$x$ $f(x)$ $g(x)$
$-2$ $1$ $\frac{1}{4}$
$-1$
$0$
$1$
$2$ $1$ $4$
Based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?
between $x = -2$ and $x = -1$
$x = -1$
between $x = -1$ and $x = 0$
$x = 0$
1. **State the problem:** We need to complete the table for $f(x) = \frac{1}{2}|x|$ and $g(x) = 2^x$ at $x = -1, 0, 1$ and determine where $f(x) = g(x)$.
2. **Calculate $f(x)$ values:**
- $f(-1) = \frac{1}{2}|-1| = \frac{1}{2} \times 1 = \frac{1}{2}$
- $f(0) = \frac{1}{2}|0| = 0$
- $f(1) = \frac{1}{2}|1| = \frac{1}{2}$
3. **Calculate $g(x)$ values:**
- $g(-1) = 2^{-1} = \frac{1}{2}$
- $g(0) = 2^0 = 1$
- $g(1) = 2^1 = 2$
4. **Complete the table:**
\begin{tabular}{c|c|c}
$x$ & $f(x)$ & $g(x)$ \\
\hline
$-2$ & $1$ & $\frac{1}{4}$ \\
$-1$ & $\frac{1}{2}$ & $\frac{1}{2}$ \\
$0$ & $0$ & $1$ \\
$1$ & $\frac{1}{2}$ & $2$ \\
$2$ & $1$ & $4$ \\
\end{tabular}
5. **Determine where $f(x) = g(x)$:**
- At $x = -1$, $f(-1) = \frac{1}{2}$ and $g(-1) = \frac{1}{2}$, so $f(x) = g(x)$.
**Final answer:** The equation $f(x) = g(x)$ has a solution at $x = -1$.