If f is decreasing and g is increasing functions such that gof exists then gof is

Options

A. an increasing function
B. a decreasing function
C. neither increasing nor decreasing
D. none of these

Answer

To solve the problem, we need to analyze the behavior of the composite function \( g(f(x)) \) given that \( f \) is a decreasing function and \( g \) is an increasing function.

### Step-by-Step Solution:

1. **Understanding the Functions**:
- We know that \( f \) is a decreasing function. This means that if \( x_1 < x_2 \), then \( f(x_1) \geq f(x_2) \).
- We know that \( g \) is an increasing function. This means that if \( t_1 < t_2 \), then \( g(t_1) \leq g(t_2) \).

2. **Let \( t = f(x) \)**:
- Since \( f \) is decreasing, as \( x \) increases, \( t = f(x) \) will decrease. Therefore, if \( x_1 < x_2 \), then \( t_1 = f(x_1) \geq f(x_2) = t_2 \).

3. **Analyzing \( g(f(x)) \)**:
- Now, we need to find \( g(f(x)) \). Since \( t = f(x) \) is decreasing, we can say that as \( x \) increases, \( t \) decreases.
- Therefore, if \( t_1 = f(x_1) \) and \( t_2 = f(x_2) \) with \( x_1 < x_2 \), we have \( t_1 \geq t_2 \).

4. **Applying the Increasing Property of \( g \)**:
- Since \( g \) is increasing, and we have \( t_1 \geq t_2 \), it follows that \( g(t_1) \leq g(t_2) \).
- This means that as \( x \) increases, \( g(f(x)) \) will also decrease.

5. **Conclusion**:
- Thus, the composite function \( g(f(x)) \) is a decreasing function.

### Final Answer:
The function \( g(f(x)) \) is a decreasing function.

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