Let `f : R to and f (x) = (x (x^(4) + 1) (x+1) +x ^(4)+2)/(x^(2) +x+1),` then `f (x)` is :

Options

A. One-one, into
B. Many one, onto
C. One-one, onto
D. Many one, into

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{x(x^4 + 1)(x + 1) + x^4 + 2}{x^2 + x + 1} \) and determine its properties, specifically whether it is one-one, many-one, onto, or into.

### Step 1: Simplify the Function
We start with the function:
\[ f(x) = \frac{x(x^4 + 1)(x + 1) + x^4 + 2}{x^2 + x + 1} \]
Let's simplify the numerator:
\[ x(x^4 + 1)(x + 1) + x^4 + 2 = x(x^5 + x^4 + x + 1) + x^4 + 2 \]
Expanding this gives:
\[ x^6 + x^5 + x^2 + x + x^4 + 2 = x^6 + x^5 + x^4 + x^2 + x + 2 \]
So, we can rewrite \( f(x) \):
\[ f(x) = \frac{x^6 + x^5 + x^4 + x^2 + x + 2}{x^2 + x + 1} \]

### Step 2: Analyze the Denominator
The denominator \( x^2 + x + 1 \) is always positive for all real \( x \) because its discriminant \( b^2 - 4ac = 1 - 4 < 0 \). Thus, it has no real roots and does not change sign.

### Step 3: Determine the Nature of the Function
To determine whether \( f(x) \) is one-one or many-one, we can analyze the behavior of the numerator \( N(x) = x^6 + x^5 + x^4 + x^2 + x + 2 \).

### Step 4: Check for Critical Points
To find critical points, we can take the derivative of \( N(x) \):
\[ N'(x) = 6x^5 + 5x^4 + 4x^3 + 2x + 1 \]
Since all coefficients are positive, \( N'(x) > 0 \) for all \( x \). This implies that \( N(x) \) is a strictly increasing function.

### Step 5: Conclusion on One-One and Many-One
Since \( N(x) \) is strictly increasing, \( f(x) \) is also strictly increasing as the denominator is always positive. Therefore, \( f(x) \) is a one-one function.

### Step 6: Determine Onto or Into
The range of \( f(x) \) must be analyzed. As \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \). However, since \( N(x) \) is always positive, \( f(x) \) will only take positive values. Thus, the range of \( f(x) \) is \( (0, \infty) \), which does not cover all real numbers.

### Final Answer
Thus, we conclude that:
- \( f(x) \) is a **one-one** function.
- The function is **into** since the range does not equal the codomain.