Question

question #1 serena used the given rational function to determine the intervals of increase and decrease. determine if serenas work is correct. if not, state what she did incorrectly. serena states the intervals of increase and decrease to be: increase: none decrease: (-∞,-1)∪(-1,2)∪(2,∞) serena stated both intervals of increase and decrease incorrectly by writing the interval of increase as decrease and the interval of decrease as increase. serena stated the interval of decrease incorrectly including the sections of the domain which are above the x - axis. serena correctly stated the intervals of increase and decrease. serena stated the interval of increase incorrectly by including the sections of the domain which are below the x - axis.

Explanation:

Step1: Analyze increasing - decreasing intervals

For a rational - function, we look at the slope of the graph. If the slope is positive, the function is increasing; if negative, decreasing. By observing the given graph, as \(x\) moves from left to right, the function never has a positive - sloped section, so the interval of increase is none.

Step2: Check the decreasing intervals

The function has a negative slope on the intervals \((-\infty,-1)\), \((-1,2)\), and \((2,\infty)\). The vertical asymptotes at \(x = - 1\) and \(x=2\) do not affect the determination of increasing and decreasing intervals.

Answer:

Serena correctly stated the intervals of increase and decrease.