To compare the functions \( f(x) \) and \( g(x) \) and find when \( f(x) > g(x) \), we'll analyze the provided values in the table:
\[ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \ \hline -2 & -1 & 3 \ -1 & \frac{3}{4} & \frac{3}{4} \ 0 & 1 & 0 \ 1 & 1 & \frac{1}{4} \ 2 & 3 & 3 \ \hline \end{array} \]
Next, we evaluate when \( f(x) \) is greater than \( g(x) \):
- At \( x = -2 \): \( f(-2) = -1 < g(-2) = 3 \) → \( f(x) < g(x) \)
- At \( x = -1 \): \( f(-1) = \frac{3}{4} = g(-1) = \frac{3}{4} \) → \( f(x) = g(x) \)
- At \( x = 0 \): \( f(0) = 1 > g(0) = 0 \) → \( f(x) > g(x) \)
- At \( x = 1 \): \( f(1) = 1 > g(1) = \frac{1}{4} \) → \( f(x) > g(x) \)
- At \( x = 2 \): \( f(2) = 3 = g(2) = 3 \) → \( f(x) = g(x) \)
Now, we evaluate the intervals:
- For \( x < -1 \) (e.g., \( x = -2 \)): \( f(x) < g(x) \).
- For \( -1 < x < 0 \): \( f(x) \) transitions from \( g(x) = \frac{3}{4} \) which is greater than \( f(-1) = \frac{3}{4} \) to \( f(0) = 1 > g(0) = 0 \).
- Between \( 0 < x < 1 \): \( f(x) > g(x) \).
- At \( x = 1 \): \( f(x) > g(x) \).
- From \( x = 1 \) to \( 2 \): \( f(x) > g(x) \) and equal at \( x = 2 \).
Thus, \( f(x) > g(x) \) holds true in the range \( 0 < x < 2 \).
The correct response from the options provided is:
-1 < x < 2