f(x) = 0.23x − 3; g(x) = 13(0.93x); a = 15; b = 50

Let's analyze the functions and the interval you provided. We have:

1. **Functions**:
   - \( f(x) = 0.23x - 3 \)
   - \( g(x) = 13(0.93^x) \)

2. **Interval**:
   - \( a = 15 \)
   - \( b = 50 \)

If you’re interested in comparing the behavior of \( f(x) \) and \( g(x) \) over the interval \([15, 50]\), we can examine aspects such as intersection points, where one function is greater than the other, or specific values within that interval.

### Key Steps
1. **Intersection Points**: Set \( f(x) = g(x) \) and solve for \( x \).
2. **Comparison on \([15, 50]\)**: Find where \( f(x) > g(x) \) and \( f(x) < g(x) \).
3. **Evaluate at Interval Boundaries**: Check \( f(a) \), \( f(b) \), \( g(a) \), and \( g(b) \).

Would you like to proceed with any of these specific analyses?

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**Questions for Further Exploration**:
1. What is the solution for \( f(x) = g(x) \) within the interval \([15, 50]\)?
2. How does the growth of \( g(x) = 13(0.93^x) \) compare to \( f(x) = 0.23x - 3 \) as \( x \) increases?
3. What are the values of \( f(x) \) and \( g(x) \) at the endpoints \( x = 15 \) and \( x = 50 \)?
4. On which parts of \([15, 50]\) is \( f(x) \) greater than \( g(x) \) and vice versa?
5. What are the maximum and minimum values of \( f(x) \) and \( g(x) \) on this interval?

**Tip**: For functions that grow or decay, like \( g(x) = 13(0.93^x) \), observe the base of the exponential term to predict long

f(x) = 0.23x − 3; g(x) = 13(0.93^x); a = 15; b = 50

Explore the behavior and comparison of the linear function f(x) = 0.23x - 3 and the exponential decay function g(x) = 13(0.93^x) over the interval [15, 50]. This problem addresses key algebraic concepts, function intersection points, and value comparisons, suitable for high school students in Grades 10-12.

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