Question

use the graph of f to determine the intervals where f is increasing and when f is decreasing
a. increasing (-2,1); decreasing (-∞, -2),(1,∞)
b. increasing (-2,0),(1,∞); decreasing (-∞, -2),(0,1)
c. increasing (0,∞); decreasing (-∞,0)
d. increasing (-2,∞); decreasing (-∞, -2)

Explanation:

To solve this, we analyze the graph's slope (increasing when rising from left to right, decreasing when falling):

Step 1: Identify Increasing Intervals

A function is increasing when, as \( x \) increases, \( f(x) \) increases (graph rises). From the graph, we check intervals:

• For \( (-\infty, -2) \): The graph falls (decreasing, not increasing).
• For \( (-2, \infty) \): Wait, no—wait, let’s re-examine. Wait, the options: Let's check each option’s increasing/decreasing intervals.

Wait, the options:

• Option D: Increasing \( (-2, \infty) \), Decreasing \( (-\infty, -2) \).

Let’s verify:

• On \( (-\infty, -2) \): As \( x \) moves from \( -\infty \) to \( -2 \), the graph falls (so \( f(x) \) decreases).
• On \( (-2, \infty) \): As \( x \) moves from \( -2 \) to \( \infty \), the graph rises (so \( f(x) \) increases).

Answer:

D. increasing \((-2, \infty)\); decreasing \((-\infty, -2)\)