Problem 40 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f: R \rightarrow R\) be a function defined by, \(f(x)=\) \(-\frac{|x|^{3}+|x|}{1+x^{2}}\), then the graph of \(f(x)\) lies in which quadrant \((s) ?\) (A) I and II (B) I and III (C) II and III (D) III and IV

Step-by-Step Solution

Answer

Quadrants II and IV (D).

Step 1: Understand the Function

The given function is defined as \( f(x) = -\frac{|x|^{3}+|x|}{1+x^{2}} \). Here, \( |x| \) represents the absolute value of \( x \). This function involves absolute values and can take different forms depending on whether \( x \) is positive, negative, or zero.

Step 2: Analyze the Behavior for \( x > 0 \)

For \( x > 0 \), \( |x| = x \). Thus the function simplifies to \( f(x) = -\frac{x^3 + x}{1 + x^2} \). Since \( x^3 + x \) is positive for \( x > 0 \), and the whole expression has a negative sign in front, \( f(x) \) will be negative.

Step 3: Analyze the Behavior for \( x < 0 \)

For \( x < 0 \), \( |x| = -x \). Thus the function becomes \( f(x) = -\frac{(-x)^3 + (-x)}{1 + x^2} = -\frac{-x^3 - x}{1 + x^2} = \frac{x^3 + x}{1 + x^2} \). Since both \( x^3 \) and \( x \) are negative, \( f(x) \) will be positive.

Step 4: Determine the Quadrants for \( x = 0 \)

At \( x = 0 \), \( f(0) = 0 \). This implies that the point lies on the origin, \((0,0)\), which is not exclusively in any particular quadrant.

Step 5: Conclude the Quadrants

For \( x > 0 \), \( f(x) < 0 \), indicating the graph lies in Quadrant IV (where \( x \) is positive and \( y \) is negative). For \( x < 0 \), \( f(x) > 0 \), indicating the graph lies in Quadrant II (where \( x \) is negative and \( y \) is positive). Thus, the graph of \( f(x) \) lies in Quadrants II and IV.

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics. It represents the distance of a number from zero on a number line, regardless of direction. For any real number \( x \), the absolute value is denoted as \( |x| \), and it is always non-negative.

• When \( x \) is positive, \( |x| = x \).
• When \( x \) is negative, \( |x| = -x \).
• When \( x = 0 \), \( |x| = 0 \).

Absolute value helps in simplifying expressions, especially in cases like piecewise functions or equations where \( x \) can change signs. In the given function, the absolute value causes the function to behave differently depending on whether \( x \) is positive or negative. This is crucial for determining the overall behavior of the function across different values of \( x \).

Quadrant Analysis

The plane of a Cartesian coordinate system is divided into four regions known as quadrants. Quadrants are essential for graphing and analyzing functions, as they help us determine where parts of the graph lie. Each quadrant has specific characteristics:

• Quadrant I: Both \( x \) and \( y \) are positive.
• Quadrant II: \( x \) is negative and \( y \) is positive.
• Quadrant III: Both \( x \) and \( y \) are negative.
• Quadrant IV: \( x \) is positive and \( y \) is negative.

In our exercise, we're determining which quadrants the graph of \( f(x) \) lies in. For \( x > 0 \), \( f(x) < 0 \), indicating Quadrant IV. For \( x < 0 \), \( f(x) > 0 \), indicating Quadrant II. This understanding guides us in placing the graph correctly on the Cartesian plane.

Graph Behavior Analysis

Analyzing the graph of a function like \( f(x) = -\frac{|x|^3 + |x|}{1 + x^2} \) involves understanding how \( f(x) \) behaves as \( x \) changes across the real number line. Graph behavior includes aspects like concavity, symmetry, and intercepts.Let's consider the specific cases:

• For \( x > 0 \): The function is negative, contributing points to Quadrant IV.


• For \( x < 0 \): The function becomes positive, contributing points to Quadrant II.


• At \( x = 0 \): The function equals zero, placing a point at the origin, though not impacting quadrant placement.

The numerator \( |x|^3 + |x| \) ensures that the positivity or negativity of the output rests on the sign of \( x \), as dictated by the absolute value. This behavior is mirrored over the y-axis due to symmetry from the absolute value term. Such analysis is crucial for accurately sketching and interpreting the graph of any function.