Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f: R \rightarrow R\) and \(g: R \rightarrow R\) are given by \(f(x)=|x|\) and \(g(x)=[x]\) for each \(x \in R\), then \(\\{x \in R: g[f(x)] \leq f\) \([g(x)]\\}=\) (A) \(Z \cup(-\infty, 0)\) (B) \((-\infty, 0)\) (C) \(Z\) (D) \(R\)

Step-by-Step Solution

Answer

(D) R

Step 1: Understand the Functions

Function \(f(x) = |x|\) represents the absolute value function. It takes any real number \(x\) and returns its non-negative value. Function \(g(x) = [x]\) is the floor function, which returns the greatest integer less than or equal to \(x\). We need to find the set where \(g(f(x)) \leq f(g(x))\).

Step 2: Evaluate \(g(f(x))\)

Since \(f(x) = |x|\), \(g(f(x)) = g(|x|)\). The floor function applied to \(|x|\) returns the largest integer less than or equal to \(|x|\).

Step 3: Evaluate \(f(g(x))\)

For \(g(x) = [x]\), \(f(g(x)) = f([x]) = |[x]|\). This is the absolute value of the largest integer less than or equal to \(x\), which is a non-negative integer.

Step 4: Analyze Inequality for Positive \(x\)

For positive \(x\), \(|x| = x\). Hence, \(g(f(x)) = [x]\) and \(f(g(x)) = |[x]| = [x]\). The inequality \([x] \leq [x]\) is always true.

Step 5: Analyze Inequality for Negative \(x\)

For negative \(x\), \(f(x) = |x| = -x\) which is positive. Therefore \(g(f(x)) = [-x]\). Because \(-x\) is positive, \(g(-x)\) is a negative or zero integer. However, \(f(g(x)) = |[x]| = |[x]|\) will be non-negative for negative \(x\), making the inequality hold.

Step 6: Analyze Inequality at Integer Points

At integer points, both functions resolve to the integer itself. For instance, if \(x = n\) where \(n\) is an integer, \(g(f(n)) = |n|\) and \(f(g(n)) = |n|\). Thus, the inequality \(n \leq n\) is true for any integer.

Step 7: Conclusion

Since the inequality holds for all real numbers, including positive, negative, and integer values, the solution set is the entire set of real numbers \(R\).

Key Concepts

Absolute Value Function

The absolute value function is a mathematical function that is often denoted by \( |x| \). This function is quite straightforward: it returns the non-negative value of any real number \( x \). For example, \( |3| = 3 \) and \( |-3| = 3 \).

It's important to note that the absolute value essentially measures the "distance" of a number from 0 on the number line,

• If \( x \) is positive, then \( |x| = x \).
• If \( x \) is zero, then \( |x| = 0 \).
• If \( x \) is negative, then \( |x| = -x \).

This function plays a critical role in understanding various mathematical contexts, especially those involving distances, as it ensures that values are treated in a "positive" sense.

In this exercise, the absolute value function transforms any input into its non-negative counterpart, which significantly impacts how we analyze inequalities.

Floor Function

The floor function, commonly represented as \( [x] \) or \( \text{floor}(x) \), is another fundamental mathematical tool. This function helps convert a real number into the greatest integer less than or equal to that number. For instance, \( [2.9] = 2 \) and \( [-2.1] = -3 \). Essentially, it "rounds down" a number to the nearest integer towards negative infinity.

Some key behaviors of the floor function are:

• For a positive non-integer \( x \), \( [x] \) is the largest integer less than \( x \).
• For a negative non-integer \( x \), \( [x] \) is the smallest integer less than or equal to \( x \).
• For any integer \( x \), \( [x] = x \).

The floor function is vital in mathematical analysis and various applications like programming, where precise control over integer values is necessary. In the context of our exercise, the floor function is used to impose restrictions by selecting only whole numbers from the absolute value calculations.

Inequalities in Mathematics

Inequalities are a fundamental aspect of mathematics, used to compare the relative size or order of two values. They are essential in solving equations and understanding the behavior of functions. Common inequality symbols include:

• \( < \) (less than)
• \( > \) (greater than)
• \( \leq \) (less than or equal to)
• \( \geq \) (greater than or equal to)

Solving inequalities often involves finding the range of values that satisfy a given condition. In this exercise, we analyzed the inequality \( g(f(x)) \leq f(g(x)) \). The aim was to find all \( x \, \) such that applying both the absolute value and floor functions would satisfy this inequality.

Understanding inequalities involves the careful consideration of how different mathematical functions interact. In our exercise, by analyzing the outputs of the functions for both positive and negative \( x \), we determined that the inequality held for all real numbers \( x \), which is a crucial step in the solution process.