Question

determine the open intervals on which the function is increasing, decreasing, or constant. (enter your answers using interval notation. if an answer does not exist, enter dne.) f(x) = √(x² - 25)

Explanation:

Step1: Analyze left - hand side of graph

The left - hand side of the graph from negative infinity to - 5 is not part of the domain (since \(y = \sqrt{x^{2}-25}\) requires \(x^{2}-25\geq0\)). From \(x=-5\) to \(x = 0\), as \(x\) increases, \(y\) increases.

Step2: Analyze right - hand side of graph

The right - hand side of the graph from \(x = 0\) to \(x=5\) is not part of the domain. From \(x = 5\) to positive infinity, as \(x\) increases, \(y\) decreases.

Step3: Determine intervals

For an increasing function, the \(y\) - value increases as the \(x\) - value increases. For a decreasing function, the \(y\) - value decreases as the \(x\) - value increases. There are no constant intervals.

Answer:

Increasing: \((-5,0)\); Decreasing: \((5,\infty)\); Constant: DNE