Question

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

Explanation:

Step1: Define decreasing function

A function is decreasing on an interval if as $x$ increases, $f(x)$ decreases. We need to check that this only happens for $x > -1$, and for $x < -1$, the function is increasing or constant.

Step2: Check Top-Left Table

$x$: -3, -2, -1, 0, 1; $f(x)$: -3, -5, -7, -6, 1

• For $x < -1$ ($x=-3\to-2$): $f(x)$ goes from -3 to -5 (decreasing)
• For $x > -1$ ($x=-1\to0\to1$): $f(x)$ goes from -7 to -6 to 1 (increasing)

This does not fit.

Step3: Check Top-Right Table

$x$: -3, -2, -1, 0, 1, 2; $f(x)$: -1, -3, -5, -2, -1, 2

• For $x < -1$ ($x=-3\to-2\to-1$): $f(x)$ goes from -1 to -3 to -5 (decreasing)
• For $x > -1$ ($x=-1\to0\to1\to2$): $f(x)$ goes from -5 to -2 to -1 to 2 (increasing)

This does not fit.

Step4: Check Middle Table

$x$: -3, -2, -1, 0, 1, 2; $f(x)$: -4, -3, -1, 2, 1, -6

• For $x < -1$ ($x=-3\to-2\to-1$): $f(x)$ goes from -4 to -3 to -1 (increasing)
• For $x > -1$ ($x=-1\to0\to1\to2$): $f(x)$ goes from -1 to 2 (increasing), then 2 to 1 to -6 (decreasing)

This does not fit (increases first for $x > -1$).

Step5: Check Bottom Table

$x$: -3, -2, -1, 0, 1, 2; $f(x)$: -5, -1, 1, 0, -4, -8

• For $x < -1$ ($x=-3\to-2\to-1$): $f(x)$ goes from -5 to -1 to 1 (increasing)
• For $x > -1$ ($x=-1\to0\to1\to2$): $f(x)$ goes from 1 to 0 to -4 to -8 (decreasing)

This fits: only decreasing over $(-1, \infty)$.

Answer:

The bottom table (with $x$ values -3, -2, -1, 0, 1, 2 and corresponding $f(x)$ values -5, -1, 1, 0, -4, -8)