if `f(g(x))` is one-one function, then

Options

A. g(x) must be one-one
B. f(x) must be one-one
C. f(x) may not be one-one
D. g(x) may not be one-one

Answer

To solve the problem, we need to analyze the implications of the function composition \( f(g(x)) \) being a one-one function.

### Step-by-Step Solution:

1. **Understanding One-One Functions**:
A function \( h(x) \) is one-one (or injective) if for any two inputs \( x_1 \) and \( x_2 \), \( h(x_1) = h(x_2) \) implies \( x_1 = x_2 \).

2. **Given Condition**:
We are given that \( f(g(x)) \) is a one-one function. This means that if \( f(g(x_1)) = f(g(x_2)) \), then it must follow that \( x_1 = x_2 \).

3. **Applying the Definition**:
From the definition of a one-one function, if \( f(g(x_1)) = f(g(x_2)) \), we can conclude:
\[ g(x_1) = g(x_2) \quad \text{(since \( f \) must map distinct inputs to distinct outputs)} \]

4. **Implication for \( g \)**:
Since \( g(x_1) = g(x_2) \) implies \( x_1 = x_2 \), we can conclude that \( g \) must also be a one-one function.

5. **Conclusion**:
Therefore, if \( f(g(x)) \) is a one-one function, it implies that \( g(x) \) must be a one-one function. The function \( f(x) \) does not necessarily have to be one-one for \( f(g(x)) \) to be one-one.

### Final Answer:
- The condition that must be true is that \( g(x) \) is a one-one function.