Question

which table shows a function that is decreasing only over the interval (-1, 1)?

Explanation:

To determine which table shows a function decreasing only over the interval \((-1, 1)\), we analyze the change in \(f(x)\) as \(x\) increases from \(-1\) to \(1\) (i.e., for \(x = -1, 0, 1\)) and check the behavior outside this interval. A function is decreasing when as \(x\) increases, \(f(x)\) decreases.

Analyze Table 1:

\(x\): \(-2, -1, 0, 1, 2\)
\(f(x)\): \(0, -3, 0, 3, 0\)

• From \(x = -2\) to \(x = -1\): \(f(x)\) decreases (from \(0\) to \(-3\)).
• From \(x = -1\) to \(x = 1\): \(f(x)\) increases (from \(-3\) to \(3\)).
• From \(x = 1\) to \(x = 2\): \(f(x)\) decreases (from \(3\) to \(0\)).

This function decreases over \((-2, -1)\) and \((1, 2)\), not only over \((-1, 1)\).

Analyze Table 2:

\(x\): \(-2, -1, 0, 1, 2\)
\(f(x)\): \(0, 3, 0, -3, 0\)

• From \(x = -2\) to \(x = -1\): \(f(x)\) increases (from \(0\) to \(3\)).
• From \(x = -1\) to \(x = 1\): \(f(x)\) decreases (from \(3\) to \(-3\)).
• From \(x = 1\) to \(x = 2\): \(f(x)\) increases (from \(-3\) to \(0\)).

This function decreases only over \((-1, 1)\) (since it increases outside this interval).

Analyze Table 3:

\(x\): \(-2, -1, 0, 1, 2\)
\(f(x)\): \(-10, -8, 0, 8, 10\)

• From \(x = -2\) to \(x = 2\): \(f(x)\) increases (consistently rising).

This function is always increasing, not decreasing over \((-1, 1)\).

Analyze Table 4:

\(x\): \(-2, -1, 0, 1, 2\)
\(f(x)\): \(10, 8, 0, -8, -10\)

• From \(x = -2\) to \(x = 2\): \(f(x)\) decreases (consistently falling).

This function is always decreasing, not only over \((-1, 1)\).

Only Table 2 shows a function that decreases only over the interval \((-1, 1)\).

Answer:

The second table (with \(x\)-values \(-2, -1, 0, 1, 2\) and \(f(x)\)-values \(0, 3, 0, -3, 0\))