Which of the following function/functions is/are periodic ?
(a) `sgn(e^(-x)) " (b) " sinx + |sinx|`
(c ) `min(sinx, |x|) " (d) " (x)/(x)`

Explanation

To determine which of the given functions is periodic, we will analyze each option step by step.

### Step 1: Analyze Option (a) \( \text{sgn}(e^{-x}) \)

The function \( \text{sgn}(e^{-x}) \) is defined as follows:
- \( \text{sgn}(y) = 1 \) if \( y > 0 \)
- \( \text{sgn}(y) = 0 \) if \( y = 0 \)
- \( \text{sgn}(y) = -1 \) if \( y < 0 \)

Since \( e^{-x} \) is always positive for all real \( x \), we have:
\[ \text{sgn}(e^{-x}) = 1 \text{ for all } x \]

This means the function is constant and does not repeat values over any interval. Therefore, it is not periodic.

### Conclusion for (a):
**Not periodic.**

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### Step 2: Analyze Option (b) \( \sin x + |\sin x| \)

The function can be rewritten as:
- If \( \sin x \geq 0 \), then \( |\sin x| = \sin x \) and \( f(x) = \sin x + \sin x = 2\sin x \)
- If \( \sin x < 0 \), then \( |\sin x| = -\sin x \) and \( f(x) = \sin x - \sin x = 0 \)

The function is periodic because:
- \( 2\sin x \) is periodic with period \( 2\pi \)
- The constant function \( 0 \) is also periodic.

Thus, the overall function \( f(x) \) retains periodicity.

### Conclusion for (b):
**Periodic with period \( 2\pi \).**

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### Step 3: Analyze Option (c) \( \min(\sin x, |x|) \)

To analyze this function, we need to consider the behavior of both \( \sin x \) and \( |x| \):
- \( \sin x \) is periodic with period \( 2\pi \).
- \( |x| \) is not periodic; it increases indefinitely as \( x \) moves away from zero.

The minimum function will switch between \( \sin x \) and \( |x| \) depending on the value of \( x \). As \( |x| \) grows larger than \( \sin x \), the function will not repeat its values, leading to a break in periodicity.

### Conclusion for (c):
**Not periodic.**

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### Step 4: Analyze Option (d) \( \frac{x}{x} \)

The function \( \frac{x}{x} \) is defined as:
- \( 1 \) for \( x \neq 0 \)
- Undefined for \( x = 0 \)

Since the function takes the constant value \( 1 \) for all \( x \) except at \( x = 0 \), it does not repeat values over any interval. Therefore, it is not periodic.

### Conclusion for (d):
**Not periodic.**

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### Final Summary:
- (a) Not periodic
- (b) Periodic
- (c) Not periodic
- (d) Not periodic