Determine the largest open intervals of the domain over which each function is (a) increasing. See Example 9.

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Identify the critical points on the graph where the function changes from increasing to decreasing or vice versa. In this graph, the critical points are at \(x = -3\) and \(x = 0\).

Recall that a function is increasing on intervals where the graph moves upward as \(x\) increases. Look at the graph to determine where the function rises as you move from left to right.

From the graph, observe that the function increases from the far left (negative infinity) up to the point \((-3, 5)\). This means the function is increasing on the interval \((-\infty, -3)\).

After \(x = -3\), the function starts to decrease until \(x = 0\). At \(x = 0\), the function becomes constant (horizontal line), so it is neither increasing nor decreasing beyond this point.

Therefore, the largest open interval where the function is increasing is \((-\infty, -3)\). Write this interval as your answer.

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Increasing and Decreasing Intervals

A function is increasing on an interval if, as x moves from left to right, the function values rise. Conversely, it is decreasing if the function values fall. Identifying these intervals involves analyzing the graph or the derivative to see where the slope is positive or negative.

Open Intervals

Open intervals are intervals that do not include their endpoints. When determining where a function is increasing, we focus on the largest open intervals where the function consistently rises, excluding points where the function changes direction or is constant.

Critical Points and Local Extrema

Critical points occur where the function's derivative is zero or undefined, often corresponding to local maxima or minima. These points mark where the function changes from increasing to decreasing or vice versa, helping to identify the boundaries of increasing intervals.