Which of the following is not an odd function ?

Options

A. ln `((x ^(4)+x^(2)+1)/((x^(2)+x+1)^(2)))`
B. sgn (sgn(x))
C. sin (tan x)
D. `f (x),` where `f (x)+ f ((1)/(x)) =f(x) f ((1)/(x))AA x in R-{0} and f(2) =33`

Answer

To determine which of the given functions is not an odd function, we need to recall the definition of an odd function. A function \( f(x) \) is considered odd if it satisfies the condition:

\[ f(-x) = -f(x) \]

Now, let's analyze each option step by step.

### Step 1: Analyze Option 1
**Function:** \( f(x) = \frac{\log(x^4 + x^2 + 1)}{(x^2 + x + 1)^2} \)

**Check:** We need to evaluate \( f(-x) \).

\[ f(-x) = \frac{\log((-x)^4 + (-x)^2 + 1)}{((-x)^2 + (-x) + 1)^2} \]

Calculating \( f(-x) \):

\[ = \frac{\log(x^4 + x^2 + 1)}{(x^2 - x + 1)^2} \]

Now, we compare \( f(-x) \) with \( -f(x) \):

\[ -f(x) = -\frac{\log(x^4 + x^2 + 1)}{(x^2 + x + 1)^2} \]

Since \( f(-x) \neq -f(x) \), this function is **not an odd function**.

### Step 2: Analyze Option 2
**Function:** \( f(x) = \sin(\sin x) \)

**Check:** Evaluate \( f(-x) \).

\[ f(-x) = \sin(\sin(-x)) = \sin(-\sin x) = -\sin(\sin x) = -f(x) \]

Since \( f(-x) = -f(x) \), this function is an **odd function**.

### Step 3: Analyze Option 3
**Function:** \( f(x) = \sin(\tan x) \)

**Check:** Evaluate \( f(-x) \).

\[ f(-x) = \sin(\tan(-x)) = \sin(-\tan x) = -\sin(\tan x) = -f(x) \]

Since \( f(-x) = -f(x) \), this function is also an **odd function**.

### Step 4: Analyze Option 4
**Function:** \( f(x) \) defined by the equation \( f(x) + f\left(\frac{1}{x}\right) = f(x) \cdot f\left(\frac{1}{x}\right) \)

**Check:** We need to evaluate \( f(-x) \).

Substituting \( -x \):

\[ f(-x) + f\left(-\frac{1}{x}\right) = f(-x) \cdot f\left(-\frac{1}{x}\right) \]

This does not directly help us show that \( f(-x) \neq -f(x) \). However, we can analyze the implications of this equation.

Assuming \( f(x) \) is odd, we would have:

\[ f(-x) + f\left(-\frac{1}{x}\right) = -f(x) - f\left(\frac{1}{x}\right) \]

This leads to contradictions based on the nature of \( f \). Thus, we conclude that this function is **not an odd function**.

### Conclusion
The functions that are not odd functions are:
1. Option 1: \( \frac{\log(x^4 + x^2 + 1)}{(x^2 + x + 1)^2} \)
2. Option 4: \( f(x) + f\left(\frac{1}{x}\right) = f(x) \cdot f\left(\frac{1}{x}\right) \)

### Final Answer
The functions that are **not odd functions** are Option 1 and Option 4.

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