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

Explanation:

Step1: Recall increasing - function definition

A function $y = f(x)$ is increasing on an interval if for any two points $x_1$ and $x_2$ in the interval with $x_1<x_2$, we have $f(x_1)<f(x_2)$. Graphically, the function is increasing when the graph rises from left - to - right.

Step2: Analyze the given graph

We look at the graph and identify the intervals where the curve moves upwards as we move from left to right. Let's assume the $x$ - axis has tick marks at regular intervals. Suppose the tick marks are at integer values of $x$. We observe that the function is increasing on the intervals where the slope of the tangent line to the curve is positive. If we assume the left - most point of interest is at $x = a$, the next turning point at $x = b$, then at $x = c$, and so on. By visual inspection, if we assume the graph has local minima and maxima at integer values of $x$ for simplicity, we see that the function is increasing on the intervals where the graph is rising. Let's say the graph has local minima at $x_1,x_3,\cdots$ and local maxima at $x_2,x_4,\cdots$. The function is increasing on the intervals $(x_1,x_2),(x_3,x_4),\cdots$. For example, if the local minimum is at $x=- 3$ and the local maximum is at $x=-1$, and another local minimum is at $x = 1$ and local maximum is at $x = 3$, the function is increasing on the intervals $(-3,-1)$ and $(1,3)$.

Answer:

The intervals on which the graph of $f$ is increasing need to be determined by visually inspecting the graph for the parts where it rises from left - to - right. Without specific $x$ - values marked on the graph, we can only say that if the local minima are at $x_{min1},x_{min2},\cdots$ and local maxima are at $x_{max1},x_{max2},\cdots$, the increasing intervals are of the form $(x_{min1},x_{max1}),(x_{min2},x_{max2}),\cdots$. If we assume integer - valued tick marks and local minimum at $x=-3$ and local maximum at $x = - 1$ and local minimum at $x = 1$ and local maximum at $x = 3$, the increasing intervals are $(-3,-1)$ and $(1,3)$.