y = f(x) be a real valued function with domain as all real numbers. If the graph of the function is symmetrical about the line x = 1, then for all `alpha in R `

Options

A. `f(alpha)=f(alpha+1)`
B. `f(alpha-1=f(alpha)`
C. `f(alpha-1)=f(alpha+1)`
D. `f(1-alpha)=f(1+alpha)`

Answer

To solve the problem, we need to analyze the symmetry of the function \( y = f(x) \) about the line \( x = 1 \).

### Step-by-Step Solution:

1. **Understanding Symmetry**:
The symmetry about the line \( x = 1 \) means that for any point on the graph of the function, if you take a point \( (1 + \alpha, f(1 + \alpha)) \), there should be a corresponding point \( (1 - \alpha, f(1 - \alpha)) \) such that both points are equidistant from the line \( x = 1 \).

2. **Setting Up the Equation**:
From the symmetry condition, we can express this mathematically as:
\[ f(1 - \alpha) = f(1 + \alpha) \]
This equation states that the function values at points equidistant from \( x = 1 \) are equal.

3. **Conclusion**:
Therefore, for all \( \alpha \in \mathbb{R} \), the relationship \( f(1 - \alpha) = f(1 + \alpha) \) must hold true.

### Final Answer:
The correct statement is:
\[ f(1 - \alpha) = f(1 + \alpha) \quad \text{for all } \alpha \in \mathbb{R} \]