if f is an increasing function and g is a decreasing function on an interval I such that fog exists then

Options

A. fog is an increasing function on I
B. fog is a decreasing function on I
C. fog is neither increasing nor decreasing on I
D. None of these

Answer

To solve the problem, we need to analyze the composition of two functions: \( f \) (an increasing function) and \( g \) (a decreasing function). We want to determine the nature of the composite function \( f(g(x)) \).

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- A function \( f(x) \) is called increasing on an interval \( I \) if for any \( x_1, x_2 \in I \) such that \( x_1 < x_2 \), we have \( f(x_1) < f(x_2) \).
- A function \( g(x) \) is called decreasing on an interval \( I \) if for any \( x_1, x_2 \in I \) such that \( x_1 < x_2 \), we have \( g(x_1) > g(x_2) \).

2. **Choosing Points in the Interval**:
- Let's consider two points \( x_1 \) and \( x_2 \) in the interval \( I \) such that \( x_1 < x_2 \).

3. **Applying the Decreasing Function**:
- Since \( g \) is decreasing, we have:
\[ g(x_1) > g(x_2) \]

4. **Applying the Increasing Function**:
- Now, since \( f \) is increasing, we can apply \( f \) to both sides of the inequality:
\[ f(g(x_1)) > f(g(x_2)) \]

5. **Conclusion**:
- The inequality \( f(g(x_1)) > f(g(x_2)) \) shows that \( f(g(x)) \) is a decreasing function because as \( x \) increases, \( f(g(x)) \) decreases. Therefore, we can conclude that the composite function \( f(g(x)) \) is decreasing on the interval \( I \).

### Final Result:
Thus, if \( f \) is an increasing function and \( g \) is a decreasing function on an interval \( I \), then the composite function \( f(g(x)) \) is a decreasing function on that interval.