what is ARC for g(x) interval (-3,1)g(x)x-15/2x

It looks like you're asking for the **Average Rate of Change (ARC)** of a function \( g(x) \) over the interval \( (-3, 1) \), with the function definition \( g(x) = \frac{x - 15}{2x} \). 

The formula for the **Average Rate of Change (ARC)** of a function \( g(x) \) over an interval \([a, b]\) is:

\[
\text{ARC} = \frac{g(b) - g(a)}{b - a}
\]

Here, we are given \( a = -3 \) and \( b = 1 \).

Let's calculate it step by step:

1. **Evaluate \( g(1) \):**

\[
g(1) = \frac{1 - 15}{2(1)} = \frac{-14}{2} = -7
\]

2. **Evaluate \( g(-3) \):**

\[
g(-3) = \frac{-3 - 15}{2(-3)} = \frac{-18}{-6} = 3
\]

3. **Find the ARC:**

\[
\text{ARC} = \frac{g(1) - g(-3)}{1 - (-3)} = \frac{-7 - 3}{1 + 3} = \frac{-10}{4} = -\frac{5}{2}
\]

So, the **Average Rate of Change** of \( g(x) \) over the interval \( (-3, 1) \) is \( -\frac{5}{2} \).

Would you like more details or have any questions?

Here are 5 related questions you might want to explore:

1. How do you interpret the average rate of change geometrically?
2. How is the ARC of a function related to its derivative?
3. What is the ARC for a quadratic function over a given interval?
4. How would the ARC change if the interval were different?
5. How do you calculate the instantaneous rate of change at a point?

**Tip:** The Average Rate of Change over an interval is similar to finding the slope of the secant line between two points on a curve!

Explore how to calculate the Average Rate of Change (ARC) for the function g(x) = (x - 15) / 2x over the interval (-3, 1). This detailed solution includes function evaluations, ARC formula, and interpretation of results. Perfect for students in Grades 10-12.

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