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

Verified step by step guidance

Step 1: Understand that a function is constant on an interval if its output value (y-value) does not change as the input (x-value) changes within that interval.

Step 2: Look at the graph and identify the intervals where the function's graph is a horizontal line, since horizontal lines indicate constant function values.

Step 3: From the graph, observe that the function is constant on the interval starting at x = 0 and continuing to the right, where the function value remains at y = -4.

Step 4: Note that the function is not constant on the interval from x = -3 to x = 5 because the graph is curved and the y-values change; however, the graph shows a point at (-3, 5) which is a single point, not an interval.

Step 5: Conclude that the largest open interval where the function is constant is from x = 0 to positive infinity, written as $(0, \infty)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Constant Function

A constant function is one where the output value remains the same for every input in a given interval. On a graph, this appears as a horizontal line segment. Identifying constant intervals involves finding where the function's value does not change as the input varies.

Open Intervals

An open interval is a range of values between two endpoints, excluding the endpoints themselves. When determining intervals where a function is constant, it is important to specify open intervals to avoid including points where the function might change behavior.

Graph Analysis for Function Behavior

Analyzing a graph helps identify intervals where the function is increasing, decreasing, or constant. By observing the flat (horizontal) portions of the graph, one can determine the largest open intervals where the function remains constant, as shown by the horizontal line starting at (0, -4).

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