If `f (x) =[ln (x)/(e)] +[ln (e)/(x)],` where [.] denotes greatest interger function, the which of the following are ture ?

Options

A. range of `f (x) is {-1,0}`
B. If `f(x) =-1,` then x can be rational as well as irrational
C. If `f(x) =-1,` then x can be rational as well as irrational
D. `f (x)` is periodic function

Answer

To solve the problem, we need to analyze the function given by:

\[ f(x) = \left\lfloor \frac{\ln(x)}{e} \right\rfloor + \left\lfloor \frac{\ln(e)}{x} \right\rfloor \]

where \(\lfloor . \rfloor\) denotes the greatest integer function.

### Step 1: Simplify the Function

We know that \(\ln(e) = 1\). Therefore, we can rewrite the function as:

\[ f(x) = \left\lfloor \frac{\ln(x)}{e} \right\rfloor + \left\lfloor \frac{1}{x} \right\rfloor \]

### Step 2: Analyze Each Term

1. **First Term:** \(\left\lfloor \frac{\ln(x)}{e} \right\rfloor\)
- This term depends on the value of \(\ln(x)\). As \(x\) increases, \(\ln(x)\) increases, and thus \(\frac{\ln(x)}{e}\) increases.

2. **Second Term:** \(\left\lfloor \frac{1}{x} \right\rfloor\)
- This term is equal to 0 for all \(x > 1\) and is 1 for \(x = 1\). For \(0 < x < 1\), this term will be 1.

### Step 3: Determine the Values of \(f(x)\)

- For \(x = 1\):
\[ f(1) = \left\lfloor \frac{\ln(1)}{e} \right\rfloor + \left\lfloor \frac{1}{1} \right\rfloor = \left\lfloor 0 \right\rfloor + \left\lfloor 1 \right\rfloor = 0 + 1 = 1 \]

- For \(x > 1\):
\[ f(x) = \left\lfloor \frac{\ln(x)}{e} \right\rfloor + 0 = \left\lfloor \frac{\ln(x)}{e} \right\rfloor \]
This can take various integer values depending on \(x\).

- For \(0 < x < 1\):
\[ f(x) = \left\lfloor \frac{\ln(x)}{e} \right\rfloor + 1 \]
Here, \(\ln(x)\) is negative, so \(\frac{\ln(x)}{e}\) is also negative, and thus \(\left\lfloor \frac{\ln(x)}{e} \right\rfloor\) will be a negative integer.

### Step 4: Determine the Range of \(f(x)\)

- As \(x\) approaches 1 from the left, \(f(x)\) approaches 0.
- As \(x\) approaches 1 from the right, \(f(x)\) can take values starting from 0 and increasing.
- Thus, the possible values of \(f(x)\) are:
- For \(x = 1\), \(f(1) = 1\).
- For \(x > 1\), \(f(x)\) can be any non-negative integer.
- For \(0 < x < 1\), \(f(x)\) can take negative integer values.

### Conclusion

The range of \(f(x)\) is \([-1, 1)\) for \(0 < x < 1\) and \(f(x) \geq 0\) for \(x \geq 1\).

### Final Answer

The true statements regarding the function \(f(x)\) can be summarized as:
1. The range of \(f(x)\) includes negative integers and 0.
2. \(f(x)\) can take the value of -1 for certain values of \(x\).
3. The function is not periodic.