Question

determine all intervals on which the graph of ( f ) is increasing.

Explanation:

Step1: Recall the definition of an increasing function

A function \(y = f(x)\) is increasing on an interval if, for any two points \(x_1\) and \(x_2\) in the interval with \(x_1

Step2: Analyze the graph

• Look at the left - hand part of the graph:
• From \(x =-\infty\) to \(x=- 8\), the function is increasing (as we move from left to right, the \(y\) - values are getting larger).
• From \(x=-5\) to \(x = 0\), the function is decreasing (as we move from left to right, the \(y\) - values are getting smaller).
• From \(x = 0\) to \(x=2\), the function is increasing (as we move from left to right, the \(y\) - values are getting larger).
• From \(x = 2\) to \(x=6\), the function is decreasing (as we move from left to right, the \(y\) - values are getting smaller).
• From \(x=6\) to \(x=\infty\), the function is increasing (as we move from left to right, the \(y\) - values are getting larger).

We can express the intervals in interval notation. The intervals where the function is increasing are \((-\infty,-8)\), \((0,2)\) and \((6,\infty)\)

Answer:

The function \(f\) is increasing on the intervals \((-\infty,-8)\), \((0,2)\) and \((6,\infty)\)