If f and g are two increasing function such that fog is defined then

Options

A. gof is an increasing functions
B. gof is a decreasing function
C. gof is neither inceasing nor decreasing
D. none of these

Answer

To solve the problem, we need to determine the nature of the composition of two increasing functions, \( f \) and \( g \), specifically whether \( f \circ g \) and \( g \circ f \) are increasing functions as well.

### Step-by-Step Solution:

1. **Understanding Increasing Functions**:
An increasing function is defined such that if \( x_1 > x_2 \), then \( f(x_1) > f(x_2) \) for function \( f \) and similarly for function \( g \).

2. **Composition of Functions**:
We need to analyze the composition \( f \circ g \) and \( g \circ f \). The notation \( f \circ g \) means \( f(g(x)) \).

3. **Analyzing \( f \circ g \)**:
- Let \( x_1 > x_2 \).
- Since \( g \) is increasing, we have \( g(x_1) > g(x_2) \).
- Now, applying \( f \) (which is also increasing) to both sides, we get:
\[ f(g(x_1)) > f(g(x_2)) \]
- This implies:
\[ f \circ g (x_1) > f \circ g (x_2) \]
- Therefore, \( f \circ g \) is an increasing function.

4. **Analyzing \( g \circ f \)**:
- Again, let \( x_1 > x_2 \).
- Since \( f \) is increasing, we have \( f(x_1) > f(x_2) \).
- Now, applying \( g \) (which is also increasing) to both sides, we get:
\[ g(f(x_1)) > g(f(x_2)) \]
- This implies:
\[ g \circ f (x_1) > g \circ f (x_2) \]
- Therefore, \( g \circ f \) is also an increasing function.

5. **Conclusion**:
Since both \( f \circ g \) and \( g \circ f \) are increasing functions, we can conclude that the composition of two increasing functions is also an increasing function.

### Final Answer:
Both \( f \circ g \) and \( g \circ f \) are increasing functions.