https://file.aiquickdraw.com/mathbot/file/1726296085204nn6tul6l.jpg

The problem asks to identify specific intervals and points where the values of \( f(x) \) and \( g(x) \) (shown in the graph) meet certain conditions, such as when one function is greater or less than the other.

The functions are as follows:
- \( f(x) \): the solid line graph.
- \( g(x) \): the dashed line graph.

Let’s break down the different conditions based on the graph:

1. **Intervals where \( g(x) < f(x) \):**
   - This is asking where the dashed line graph \( g(x) \) is below the solid line graph \( f(x) \).
   - By inspecting the graph, \( g(x) \) is below \( f(x) \) in the interval:
     \[
     (-\infty, -3) \cup (7, 25)
     \]
   - These are the intervals where the dashed line lies below the solid line.

2. **\( g(x) = f(x) \) when \( x = -6, 6 \):**
   - This asks to find the points where \( g(x) \) equals \( f(x) \), i.e., where the solid and dashed lines intersect.
   - Based on the graph, the functions intersect at \( x = -6 \) and \( x = 6 \).

3. **\( g(x) > f(x) \) on the interval \( (-3, 7) \):**
   - This asks where the dashed line \( g(x) \) is above the solid line \( f(x) \).
   - The graph shows that between \( x = -3 \) and \( x = 7 \), the dashed line is indeed above the solid line.

4. **Intervals where \( f(x) < 0 \) on the interval \( -7 \leq x \leq 10 \):**
   - This asks where the solid line \( f(x) \) (solid graph) is below the x-axis (negative values).
   - Inspecting the graph, we see that \( f(x) \) is negative for the interval:
     \[
     (-5, 3.14)
     \]
   - This means the solid line is below the x-axis between these values of \( x \).

Let me know if you want further clarifications or details. 

Here are 5 related questions you could explore:
1. What happens if we find intervals where both \( f(x) \) and \( g(x) \) are positive?
2. How would the intervals change if we were looking for \( g(x) \leq f(x) \)?
3. Can we generalize how to find intersections of two graphs in more complex functions?
4. What transformations would shift \( g(x) \) above \( f(x) \) in all intervals?
5. Can we determine the areas under \( f(x) \) and \( g(x) \) between given intervals?

**Tip:** Always sketch the graph yourself to visually check where one function is above or below the other to avoid mistakes in reading the graph.

Identify the intervals and points where the graphs of f(x) and g(x) meet certain conditions based on the graph provided.

Analyze the graph of f(x) and g(x) to find intervals where one function is greater than the other, where they intersect, and where f(x) is negative. Suitable for high school students in Grades 9-11.

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