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

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

The solution set is (A) \(Z \cup (-\infty, 0)\).

Step 1: Understand the Functions

We have two functions: \(f(x) = |x|\), which is the absolute value function, and \(g(x) = [x]\), which is the floor function, mapping any real number \(x\) to the greatest integer less than or equal to \(x\).

Step 2: Evaluate g(f(x))

Since \(f(x) = |x|\), the expression \(g(f(x))\) becomes \(g(|x|)\). This is equal to \([|x|]\), which is the greatest integer less than or equal to \(|x|\).

Step 3: Evaluate f(g(x))

Using \(g(x) = [x]\), the expression \(f(g(x))\) becomes \(f([x])\), which results in \(|[x]|\). This is the absolute value of the greatest integer less than or equal to \(x\).

Step 4: Compare g(f(x)) with f(g(x))

The inequality is \(g(f(x)) \leq f(g(x))\), which translates to \([|x|] \leq |[x]|\). We check the possible range of \(x\) for which this holds.

Step 5: Analyze Outcomes for x ≥ 0

If \(x ≥ 0\), both \(|x|\) and \([x]\) are non-negative. Hence, \([x] = x\) if \(x\) is an integer, and \([x] \leq [x]\) generally holds, satisfying the inequality regardless of the decimal part.

Step 6: Analyze Outcomes for x < 0

If \(x < 0\), \(|x| = -x\), a positive value, and \( [x] \leq 0 \). Thus, \([|x|] = |-x| - 1\), not satisfying \([-x|] ≯ ||[x]||\) for any non-zero \(x\), because \(|[x]| = |-[x]| \ge 0\) emphasizing the need for \(x < 0\).

Step 7: Determining the Answer Set

The valid solution set needs to satisfy \([|x|] \leq ||[x]|| \), which holds true primarily when \(x\) is negative or \(x\) is an integer. Thus, \({x \in R : g(f(x)) \leq f(g(x))}\) encompasses integers and negative reals.

Key Concepts

Absolute Value Function

The absolute value function, denoted as \( f(x) = |x| \), is a function that transforms a real number \( x \) into its non-negative counterpart. Essentially, absolute value tells us how far a number is from zero on the number line, without considering the direction. For instance, both \(|3|\) and \(|-3|\) equal 3 because both are 3 units away from zero.
Key properties of the absolute value function include:

• \( |x| \geq 0 \) for all \( x \in R \)
• \( |x| = 0 \) if and only if \( x = 0 \)
• \( |xy| = |x||y| \) for any real numbers \( x \) and \( y \)
• \( |x+y| \leq |x| + |y| \) (triangle inequality)

In the context of the given exercise, the absolute value function is applied to \( g(x) \) to obtain \( f([x]) \). This operation ensures that we are examining the non-negative value of the greatest integer less than or equal to \( x \).
Understanding this function helps in analyzing inequalities, as it helps in estimating only positive outputs, irrespective of whether the input \( x \) is positive or negative.

Floor Function

The floor function, denoted \( g(x) = [x] \), maps a real number \( x \) to the greatest integer less than or equal to \( x \). This function can be seen as 'rounding down' any real number to an integer.
Let's consider some examples:

• \([3.7] = 3\)
• \([ -1.2 ] = -2\)
• \([5] = 5\) (as the number is already an integer)

The floor function is important for understanding how certain operations can change the size or direction of an input number. It simplifies the quantities and allows us to work with integers rather than real numbers. In the exercise, it's used to assess \( g(|x|) \), or the greatest integer less than or equal to the absolute value of \( x \).
The emphasis on integer results makes the floor function crucial for evaluating the comparison between \( g(f(x)) \) and \( f(g(x)) \). This can determine whether integer values or specific inequalities are satisfied.

Inequality Solutions

Solving inequalities involves finding the range of values within which the inequalities hold true. In the exercise, we're comparing \( g(f(x)) \) and \( f(g(x)) \) to establish \([|x|] \leq |[x]|\).
To solve this inequality, consider the following:

• When \( x \geq 0 \), \( |x| = x \), and thus \([|x|] = [x]\) which maintains equality.
• For \( x < 0 \), the function \( |x| = -x \), and the inequality \([|x|] \leq |[x]|\) means we require \([-|x|] \leq |[x]|\).
• When \( x \) is a negative integer, \([|x|] = |-x| \), and since \( x \) already rounds down, the requirement holds well for any \( x < 0 \), or when \( x \) is an integer.

The inequality solutions will therefore include all integers and negative real numbers as they satisfy \([|x|] \leq |[x]|\). By working through these steps logically, we can identify the solution set without ambiguity, ensuring a comprehensive understanding for solving mathematical inequalities involving these functions.