Problem 6 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The range of the function \(f(x)=\frac{x}{1+|x|}, x \in R\), is [Online May 7, 2012] (a) \(R\) (b) \((-1,1)\) (c) \(R-\\{0\\}\) (d) \([-1,1]\)

Step-by-Step Solution

Answer

The range of the function is \((-1, 1)\).

Step 1: Understand the Domain of the Function

The function \( f(x) = \frac{x}{1+|x|} \) is defined for all real numbers \( x \), since \( |x| \) ensures the denominator \( 1 + |x| \) is always positive and non-zero. This means the domain of the function is \( \mathbb{R} \).

Step 2: Analyze the Behavior of the Function

Examine the function as \( x \) approaches positive and negative infinity: - As \( x \to +\infty \), \( f(x) = \frac{x}{1+x} \approx 1 \). - As \( x \to -\infty \), \( f(x) = \frac{x}{1-x} \approx -1 \). This suggests the function approaches but never reaches \( 1 \) or \( -1 \).

Step 3: Check the Function at Specific Points

Evaluate the function at some key points to determine the range boundaries:- At \( x = 0 \), \( f(0) = \frac{0}{1} = 0 \).- At \( x = 1 \), \( f(1) = \frac{1}{2} \).- At \( x = -1 \), \( f(-1) = \frac{-1}{2} \).These points indicate that the function can take any value between -1 and 1, but never includes -1 or 1 itself.

Step 4: Conclusion on the Range

Based on the behavior at infinity and specific points, we deduce the function's range. The function approaches but never reaches \( 1 \) or \( -1 \). Thus, the range of the function is \( (-1, 1) \).

Key Concepts

Domain of a Function

The domain of a function represents all the possible input values that will not cause the function to break. In mathematical terms, it's a set of all real numbers for which the function is defined. For example, in the function \( f(x) = \frac{x}{1+|x|} \), we want to ensure the denominator is never zero. The absolute value function \(|x|\) comes into play here, which keeps \(1 + |x|\) strictly positive for all real numbers \(x\).

• There are no values of \(x\) that make \(1 + |x|\) equal to zero.
• This implies the function is defined for all \(x \in \mathbb{R}\).

The domain of this function is therefore the set of all real numbers \(\mathbb{R}\). This means the function can take inputs from anywhere on the number line without breaking down.

Absolute Value Function

The absolute value function \(|x|\) is a very popular function in mathematics, often used due to its ability to "strip" a number of its sign, consistently giving a non-negative result.

• The outcome of \(|x|\) is always zero or a positive number, regardless of whether \(x\) itself is negative or positive.
• This makes \(1 + |x|\) a cornerstone in ensuring the function \( f(x) = \frac{x}{1+|x|} \) remains valid for all \(x\).

Here's how it works: - If \(x > 0\), then \(|x| = x\).- If \(x < 0\), then \(|x| = -x\).This characteristic allows \(|x|\) to adapt dynamically, aiding in the consistent, smooth behavior of the function. In the context of our function, it ensures the denominator is always non-zero, making the function's domain comprehensive.

Function behavior at infinity

When exploring the behavior of functions, especially as they stretch towards infinity, it's vital to understand how the function behaves at extreme values, like when \(x\) becomes very large or very small. For the function \( f(x) = \frac{x}{1+|x|} \), let's consider both positive and negative infinity:

• As \( x \to +\infty \), the function behaves as \( f(x) = \frac{x}{1+x} \approx 1 \). This means, as \(x\) grows larger in the positive direction, the function's value approaches 1 but never actually reaches or exceeds it.
• As \( x \to -\infty \), we observe \( f(x) = \frac{x}{1-x} \approx -1 \). Here too, as \(x\) grows larger in the negative direction, the function nears -1, but never actually gets there.

This concept is particularly useful when determining the range. Since neither 1 nor -1 is reached, these values act as asymptotes, indirectly hinting at the range the function can achieve. In this instance, the values between \(-1\) and \(1\) are possible outcomes for \(f(x)\), defining its range as \((-1, 1)\).