Question

using only the values given in the table for the function, \\(f(x) = x^3 - 3x - 2\\), what is the interval of \\(x\\)-values over which the function is decreasing?

\\((-4, 1)\\)
\\((-4, -1)\\)
\\((-1, 1)\\)
\\((-1, 2)\\)

Explanation:

Analyze the definition of a decreasing function

A function \(f(x)\) is decreasing on an interval if, as the input \(x\) increases, the output \(f(x)\) decreases. Using the Function Graph Analysis knowledge point, we look for consecutive \(x\)-values where the corresponding \(f(x)\) values strictly decrease.

Examine the given table values

Let's trace the changes in \(f(x)\) as \(x\) increases from \(-4\) to \(3\):

• From \(x = -4\) to \(x = -3\): \(f(x)\) goes from \(-54\) to \(-20\) (increases)
• From \(x = -3\) to \(x = -2\): \(f(x)\) goes from \(-20\) to \(-4\) (increases)
• From \(x = -2\) to \(x = -1\): \(f(x)\) goes from \(-4\) to \(0\) (increases)
• From \(x = -1\) to \(x = 0\): \(f(x)\) goes from \(0\) to \(-2\) (decreases)
• From \(x = 0\) to \(x = 1\): \(f(x)\) goes from \(-2\) to \(-4\) (decreases)
• From \(x = 1\) to \(x = 2\): \(f(x)\) goes from \(-4\) to \(0\) (increases)
• From \(x = 2\) to \(x = 3\): \(f(x)\) goes from \(0\) to \(16\) (increases)

Determine the decreasing interval

Using the Function Graph Analysis knowledge point

$$
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$$

Thus, the function is decreasing over the interval of \(x\)-values from \(-1\) to \(1\), which is written in interval notation as \((-1, 1)\).

Answer:

• (A) \((-4, 1)\)
• (B) \((-4, -1)\)
• (C) \((-1, 1)\) (Correct answer)
• (D) \((-1, 2)\)