the function f is graphed below. determine the intervals on which f increasing and decreasing.

Explanation:

Step1: Recall increasing - decreasing rules

A function is increasing when the $y$ - values increase as the $x$ - values increase, and decreasing when the $y$ - values decrease as the $x$ - values increase.

Step2: Analyze segment 1

For the first segment (left - most part), as $x$ increases, $y$ increases. Let's assume the left - most $x$ value is $x = a$ and the right - most $x$ value of this segment is $x=- 1$. So it is increasing on $(a, - 1)$.

Step3: Analyze segment 2

For the second segment, as $x$ increases from $-1$ to $2$, $y$ increases. So it is increasing on $(-1,2)$.

Step4: Analyze segment 3

For the third segment, as $x$ increases from $2$ to $7$, $y$ decreases. So it is decreasing on $(2,7)$.

Answer:

The function $f$ is increasing on $(a, - 1)\cup(-1,2)$ and decreasing on $(2,7)$ (where $a$ is the left - most $x$ value of the first segment shown on the graph).