From the image, you have a function graph with specific points marked, and the task asks for several characteristics of this function.

Let's break down what we can infer and correct:

1. Relative maximum:

   • The relative maximum is 7 at $x = 8$, which seems to be marked correctly.

2. Relative minimum:

   • The relative minimum is -1 at $x = 0$, and this is also correct.

3. Function increasing/decreasing intervals:

   • The function is increasing on intervals where the graph goes upward.

     • From the graph, the function starts increasing after the minimum point (which occurs at $x = 0$) until it reaches its maximum at $x = 8$.
     • So, the function is increasing on the interval: $(0, 8)$.
     • The entered answer of $(-\infty, 0)\cup(8, \infty)$ is incorrect because it is not increasing from negative infinity to zero nor after $x = 8$. The correct increasing interval should just be $(0, 8)$.

   • The function is decreasing where the graph slopes downward.

     • It starts decreasing from $x = -\infty$ to the minimum at $x = 0$. Hence, the function is decreasing on the interval $(-\infty, 0)$.

4. Domain of the function:

   • The domain is the set of all possible $x$-values that the function can take.
   • Since this is a continuous function without breaks, and the graph seems to extend infinitely in both directions horizontally, the domain is $(-\infty, \infty)$.

5. Range of the function:

   • The range is the set of all possible $y$-values the function takes.
   • The minimum value of the function is $-1$ (at $x = 0$), and the maximum value is $7$ (at $x = 8$).
   • Therefore, the range of the function is $[-1, 7]$.

Corrections to be entered:

• Increasing interval: $(0, 8)$
• Domain: $(-\infty, \infty)$
• Range: $[-1, 7]$

Would you like a detailed explanation for any of these concepts?

Here are five follow-up questions for practice:

1. What is the difference between absolute maximum/minimum and relative maximum/minimum?
2. How do you determine the intervals of increasing and decreasing on a graph?
3. What changes in the graph indicate a change from increasing to decreasing, or vice versa?
4. How does the behavior of the function at its endpoints influence its range?
5. Can a function have more than one relative maximum or minimum?

Tip:

Always look for critical points on the graph where the slope changes sign (positive to negative or negative to positive) to determine maximum and minimum points.