https://file.aiquickdraw.com/mathbot/file/172550618617308nr7l7a.jpg

Let's go through the image step-by-step to address the questions.

### Information from the image:

We are given two functions:
- \( f(x) \), represented in red.
- \( g(x) \), represented in blue.

### Let's address the sub-questions:

**(a) What is \( g(2) \)?**

From the graph, when \( x = 2 \), we can find the value of \( g(x) \) (blue curve).
- From the blue graph, at \( x = 2 \), the value of \( g(2) \) is approximately \( 1 \).

Thus, \( g(2) = 1 \).

---

**(b) Find all solutions to the equation \( f(x) = -1 \).**

We need to find where the red graph intersects \( y = -1 \).
- From the graph, \( f(x) = -1 \) at \( x = 4 \) and \( x = 5 \).

Thus, the solutions are \( x = 4 \) and \( x = 5 \).

---

**(c) Where does the line \( x = 5 \) intersect the graph of \( f \)?**

The vertical line \( x = 5 \) intersects the red graph (function \( f \)) at the point where \( f(x) = -1 \).

Thus, the point of intersection is \( (5, -1) \).

---

**(d) Find the open intervals on which \( g \) is increasing/decreasing.**

- \( g(x) \) (blue graph) is increasing on the interval \( (0, 2) \).
- \( g(x) \) is decreasing on the interval \( (2, 6) \).

---

**(e) Find the average rate of change of the function \( g \) on the interval \( [-4, 2] \).**

The formula for the average rate of change of a function \( g \) on an interval \( [a, b] \) is:

\[
\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}
\]

Here, \( a = -4 \) and \( b = 2 \). From the graph:
- \( g(-4) = 0 \) (the value of \( g(x) \) at \( x = -4 \)),
- \( g(2) = 1 \).

Thus, the average rate of change is:

\[
\frac{g(2) - g(-4)}{2 - (-4)} = \frac{1 - 0}{2 + 4} = \frac{1}{6}.
\]

So, the average rate of change is \( \frac{1}{6} \).

---

### Do you want any further details or have questions on any of the parts?

Here are 5 related questions to think about:
1. What is the domain and range of the function \( g(x) \) based on the graph?
2. Find the x-intercepts of the graph of \( f(x) \).
3. At which points do \( f(x) \) and \( g(x) \) intersect?
4. What are the maximum and minimum points of the function \( f(x) \)?
5. How can you determine if a function is concave up or concave down from a graph?

#### Tip:
To determine intervals of increase or decrease for any function, focus on the slope: increasing intervals occur where the function slopes upwards (positive slope), and decreasing intervals occur where it slopes downwards (negative slope).

Consider the graphs of y = f(x) (in red) and y = g(x) (in blue). a) What is g(2)? b) Find all solutions to the equation f(x) = -1. c) Where does the line x = 5 intersect the graph of f? d) Find the open intervals on which g is increasing/decreasing. e) Find the average rate of change of the function g on the interval [-4, 2].

Explore how to analyze the graphs of functions f(x) and g(x), find where the functions intersect, determine intervals of increase and decrease, and calculate the average rate of change for the function g(x) on a specific interval. Suitable for Grades 9-11 students.

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