Which of the following functions is non-periodic?

Options

A. `f(x)=tan (3x-2)`
B. `f(x)={x},` (where {.} denotes the fractional part of x)
C. `f(x)=x+cos x`
D. `f(x)=1-(cos^(2)x)/(1+tanx)-(sin^(2)x)/(1+cot x)`

Answer

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

### Step 1: Analyze Option 1
**Function:** \( f(x) = \tan(3x - 2) \)

- The tangent function, \( \tan(x) \), is periodic with a period of \( \pi \).
- The function \( \tan(3x - 2) \) will also be periodic because the transformation does not affect the periodic nature.
- The period of \( \tan(kx) \) is given by \( \frac{\pi}{k} \). Here, \( k = 3 \), so the period is \( \frac{\pi}{3} \).

**Conclusion:** This function is periodic.

### Step 2: Analyze Option 2
**Function:** \( f(x) = \{x\} \) (where \( \{x\} \) denotes the fractional part of \( x \))

- The fractional part function \( \{x\} = x - \lfloor x \rfloor \) gives the decimal part of \( x \).
- The fractional part is always between 0 and 1, and it resets every time \( x \) increases by 1.
- Thus, the function is not periodic because it does not repeat its values in a regular interval; it resets but does not have a fixed period.

**Conclusion:** This function is non-periodic.

### Step 3: Analyze Option 3
**Function:** \( f(x) = 1 - \cos^2(x) \)

- We can simplify this as \( f(x) = \sin^2(x) \).
- The sine function \( \sin(x) \) is periodic with a period of \( 2\pi \), and thus \( \sin^2(x) \) is also periodic with a period of \( \pi \).

**Conclusion:** This function is periodic.

### Step 4: Analyze Option 4
**Function:** \( f(x) = \frac{1 - \cos^2(x)}{1 + \tan(x) - \sin^2(x)/(1 + \cot(x))} \)

- After simplification, we find that this function also involves sine and cosine, which are periodic functions.
- Therefore, the resulting function is periodic.

**Conclusion:** This function is periodic.

### Final Conclusion
From the analysis:
- Option 1: Periodic
- Option 2: **Non-periodic** (correct answer)
- Option 3: Periodic
- Option 4: Periodic

Thus, the non-periodic function among the options is **Option 2**.

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