Question

Compare the functions f(x) and g(x) from the table. When is f(x) greater than g(x)?

x f(x) g(x)
-2 -1 3
-1 3/4 3/4
0 1 0
1 1 1/4 3/4
2 3 3

Responses

-2 < x < -1

-1 > x > 2

-2 < x < 2

-1 < x < 2

Answer 1

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:

1. For \( x < -1 \) (e.g., \( x = -2 \)): \( f(x) < g(x) \).
2. For \( -1 < x < 0 \): \( f(x) \) transitions from \( g(x) = \frac{3}{4} \) which is greater than \( f(-1) = \frac{3}{4} \) to \( f(0) = 1 > g(0) = 0 \).
3. Between \( 0 < x < 1 \): \( f(x) > g(x) \).
4. At \( x = 1 \): \( f(x) > g(x) \).
5. 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**