The graph of the function `f:R rarrR` defined by `f(x)=(|x|^(2)+|x|)/(1+x^(2))` may lie in :

Options

A. `1^("st")` and `II^("nd")` quadrant
B. `II^("st")` and `III^("nd")` quadrant
C. `III^("st")` and `IV^("nd")` quadrant
D. none of these

Answer

To determine the quadrants in which the graph of the function \( f(x) = \frac{|x|^2 + |x|}{1 + x^2} \) lies, we will analyze the function step by step.

### Step 1: Analyze the Function
The function is given as:
\[ f(x) = \frac{|x|^2 + |x|}{1 + x^2} \]

### Step 2: Simplify the Function
First, we note that \( |x|^2 = x^2 \) for any real number \( x \). Thus, we can rewrite the function as:
\[ f(x) = \frac{x^2 + |x|}{1 + x^2} \]

### Step 3: Analyze the Numerator
The numerator \( x^2 + |x| \):
- If \( x \geq 0 \), then \( |x| = x \) and the numerator becomes \( x^2 + x \), which is always non-negative.
- If \( x < 0 \), then \( |x| = -x \) and the numerator becomes \( x^2 - x \), which is also non-negative since \( x^2 \) is always non-negative and \( -x \) is positive.

### Step 4: Analyze the Denominator
The denominator \( 1 + x^2 \):
- This is always positive for all real \( x \) since \( x^2 \geq 0 \) and thus \( 1 + x^2 \geq 1 \).

### Step 5: Determine the Sign of \( f(x) \)
Since both the numerator and denominator are always non-negative (and the denominator is always positive), it follows that:
\[ f(x) \geq 0 \quad \text{for all } x \in \mathbb{R} \]

### Step 6: Identify the Quadrants
The function \( f(x) \) is non-negative, meaning that the graph of \( f(x) \) will lie above the x-axis. Therefore, the graph will be located in:
- The first quadrant (where both \( x \) and \( y \) are positive).
- The second quadrant (where \( x \) is negative and \( y \) is positive).

### Conclusion
Thus, the graph of the function \( f(x) \) may lie in the first and second quadrants.

### Final Answer
The graph of the function \( f(x) \) may lie in the **first and second quadrants**.

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