I see that you've uploaded an image with a mathematical graph and several questions related to it. Let me analyze the image and solve the problems listed based on the graph provided.

Questions in the image:

6. Let $g(x)$ be the function graphed below with domain $-10 \leq x \leq 6$. Use the graph to answer the following questions:

(a) What is the range of $g$? (b) Where does $g(x) = 0$? (c) Where is the function increasing, and where is it decreasing? (d) What is the highest value of $g$? (e) Approximate $g(-9)$. (f) For what x is $g(x) = 5$? (g) Approximate $g(5)$. (h) For approximately what value(s) is $g'(x) = 0$?

I'll work through these questions using the graph and provide solutions one by one.

(a) Range of $g(x)$:

The range is the set of all possible y-values. From the graph, the lowest point of the curve seems to be around $y = -3$ and the highest point is $y = 6$. Therefore, the range is approximately: $\text{Range of } g(x) = [-3, 6]$

(b) Where does $g(x) = 0$?

$g(x) = 0$ refers to the x-values where the curve crosses the x-axis. From the graph, it appears that $g(x) = 0$ at: $x = -8, x = -3, x = 3$

(c) Where is the function increasing and decreasing?

• Increasing: A function is increasing when its slope is positive (when the curve goes upwards). From the graph, $g(x)$ increases approximately on the intervals: $(-10, -7) \quad \text{and} \quad (-3, 0)$
• Decreasing: A function is decreasing when its slope is negative (when the curve goes downwards). The decreasing intervals are: $(-7, -3) \quad \text{and} \quad (0, 6)$

(d) What is the highest value of $g(x)$?

The highest value of $g(x)$ appears to be around 6, which occurs at $x \approx -7$.

(e) Approximate $g(-9)$:

From the graph, the value of $g(-9)$ appears to be around 3.

(f) For what x is $g(x) = 5$?

Looking at the graph, $g(x) = 5$ occurs at approximately: $x \approx -9 \quad \text{and} \quad x \approx -5$

(g) Approximate $g(5)$:

From the graph, $g(5)$ appears to be around $-1$.

(h) For approximately what value(s) is $g'(x) = 0$?

$g'(x) = 0$ refers to where the slope of the curve is zero, i.e., the peaks and troughs (where the function changes direction). From the graph, these points are approximately: $x \approx -7 \quad \text{and} \quad x \approx 0$

Would you like more details on any of the solutions? Here are 5 related questions you could consider:

1. What is the domain of the function $g(x)$?
2. How do you compute the derivative $g'(x)$ from a graph?
3. What is the significance of $g'(x) = 0$ in calculus?
4. How would you find the local minima and maxima of $g(x)$?
5. Can you explain how to identify intervals of concavity for the function?

Tip: When analyzing graphs, always check for critical points such as where the function reaches a maximum or minimum, and where it crosses the axes for key insights.