Let `f:R to R ` be a function defined by `f(x)=(|x|^(3)+|x|)/(1+x^(2))`, then the graph of f(x) lies in the

Options

A. I and II quadrants
B. I and III quadrants
C. II and III quadrants
D. IIII and IV quadrants

Answer

To determine the quadrants in which the graph of the function \( f(x) = \frac{|x|^3 + |x|}{1 + x^2} \) lies, we need to analyze the behavior of the function for both positive and negative values of \( x \).

### Step 1: Analyze the function for positive \( x \)

When \( x \) is positive:
- \( |x| = x \)
- Therefore, \( f(x) = \frac{x^3 + x}{1 + x^2} \)

Since both \( x^3 \) and \( x \) are positive for \( x > 0 \), the numerator \( x^3 + x \) is positive. The denominator \( 1 + x^2 \) is also positive for all \( x \). Thus, \( f(x) > 0 \) for \( x > 0 \).

### Step 2: Analyze the function for negative \( x \)

When \( x \) is negative:
- \( |x| = -x \)
- Therefore, \( f(x) = \frac{(-x)^3 + (-x)}{1 + x^2} = \frac{-x^3 - x}{1 + x^2} \)

Here, \( -x^3 \) is positive (since \( x \) is negative) and \( -x \) is also positive. Thus, the numerator \( -x^3 - x \) is positive. The denominator \( 1 + x^2 \) remains positive. Hence, \( f(x) > 0 \) for \( x < 0 \).

### Step 3: Conclusion about the graph

From the analysis:
- For \( x > 0 \), \( f(x) > 0 \) (which corresponds to the first quadrant).
- For \( x < 0 \), \( f(x) > 0 \) (which corresponds to the second quadrant).

Thus, the graph of \( f(x) \) lies in the first and second quadrants.

### Final Answer

The graph of \( f(x) \) lies in the **first and second quadrants**.

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