Question

1. use the following graph to find the following.

(a) domain and range
(b) intervals of increase and decrease
(c) local extrema

Explanation:

Step1: Identify domain from x - values

The graph extends from \(x=-2\) to \(x = 3\). So the domain is \([-2,3]\).

Step2: Identify range from y - values

The lowest \(y\) - value is approximately \(y=-3\) and the highest is \(y = 7\). So the range is \([-3,7]\).

Step3: Determine intervals of increase

The function is increasing where the graph goes up as we move from left to right. This is on the intervals \([-2,0]\) and \([2,3]\).

Step4: Determine intervals of decrease

The function is decreasing where the graph goes down as we move from left to right. This is on the interval \([0,2]\).

Step5: Find local maxima

Local maxima occur where the function changes from increasing to decreasing. At \(x = 0\), \(y = 2\) is a local maximum.

Step6: Find local minima

Local minima occur where the function changes from decreasing to increasing. At \(x=2\), \(y=-3\) is a local minimum.

Answer:

(a) Domain: \([-2,3]\), Range: \([-3,7]\)
(b) Intervals of increase: \([-2,0]\cup[2,3]\), Intervals of decrease: \([0,2]\)
(c) Local maximum: \((0,2)\), Local minimum: \((2,-3)\)