If y = f(x) be a monotonically increasing or decreasing function of x and M is the median of variable x, then the median of y is

Options

A. `f(M)`
B. `M//2`
C. `f^(-1)(M)`
D. none of these

Answer

To solve the problem, we need to understand the relationship between the median of the variable \( x \) and the median of the transformed variable \( y = f(x) \), where \( f \) is a monotonically increasing or decreasing function.

### Step-by-Step Solution:

1. **Understanding the Median**:
The median of a dataset is the value that separates the higher half from the lower half. For a sorted list of numbers, if the number of observations \( n \) is odd, the median is the middle number. If \( n \) is even, it is the average of the two middle numbers.

2. **Identifying the Median of \( x \)**:
Let \( M \) be the median of the variable \( x \). When the values of \( x \) are arranged in ascending order, we have:
\[ x_1 < x_2 < x_3 < ... < x_n \]
If \( n \) is odd, \( M = x_{(n+1)/2} \). If \( n \) is even, \( M \) is the average of \( x_{n/2} \) and \( x_{(n/2) + 1} \).

3. **Applying the Monotonic Function**:
Since \( y = f(x) \) is a monotonically increasing or decreasing function, the order of the values will be preserved. This means:
- If \( f \) is increasing, \( f(x_1) < f(x_2) < ... < f(x_n) \).
- If \( f \) is decreasing, \( f(x_1) > f(x_2) > ... > f(x_n) \).

4. **Finding the Median of \( y \)**:
- For an increasing function, the median of \( y \) will be \( f(M) \) because the median \( M \) corresponds to the middle value in the ordered set of \( x \) values.
- For a decreasing function, the median of \( y \) will still be \( f(M) \) because the transformation does not change the position of \( M \) in the ordered set of \( y \) values.

5. **Conclusion**:
Therefore, regardless of whether \( f \) is increasing or decreasing, the median of \( y \) is given by:
\[ \text{Median of } y = f(M) \]

### Final Answer:
The median of \( y \) is \( f(M) \).

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