Problem 49 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f(x)=[x]^{2}+[x+1]-3\), where \([x]\) is greatest integer less than or equal to \(x\), then (A) \(f(x)\) is a many one and into function (B) \(f(x)=0\) for infinite number of values of \(x\) (C) \(f(x)=0\) for only two real values (D) None of these

Step-by-Step Solution

Answer

(B) \(f(x)=0\) for infinite number of values of \(x\).

Step 1: Understand the Floor Function

In the function \(f(x) = [x]^{2} + [x+1] - 3\), \([x]\) denotes the greatest integer less than or equal to \(x\). Let's denote \(n = [x]\), then \([x+1] = n+1\) if \(x\) is not an integer, and \([x+1] = n\) if \(x\) is an integer.

Step 2: Rewrite the Function

Substitute \([x] = n\) into \(f(x)\). If \(x\) is not an integer, \(f(x) = n^2 + (n + 1) - 3 = n^2 + n - 2\). If \(x\) is an integer, then \(f(x) = n^2 + n - 3\). These represent two cases.

Step 3: Analyze Function for Non-Integers

Consider \(x\) values such that \(x\) is not an integer, which means \([x] eq x\). Here, \(f(x) = n^2 + n - 2 = 0\). Solving, \[ n^2 + n - 2 = 0 \] gives roots as \(n = 1\) and \(n = -2\).

Step 4: Analyze Function for Integers

For integer \(x\), the function simplifies to \(f(x) = n^2 + n - 3\). Solve \[ n^2 + n - 3 = 0 \] for \(n\), which gives roots \(n = \frac{-3 \pm \sqrt{13}}{2}\). As \(n\) must be an integer, check if these roots yield any integer solutions—they do not.

Step 5: Check for Infinite Zeros

Since \(f(x) = n^2 + n - 2\) has integer solutions \(n = 1\) and \(n = -2\) for non-integer \(x\), values like \(x = 1.5, 1.6, 1.9, \ldots\) (which rounds to 1) and \(x = -1.5, -1.6, -1.9, \ldots\) (which rounds to -2) can make \(f(x) = 0\). This happens for infinitely many \(x\).

Step 6: Final Conclusion

Based on the analysis, choice (B) is correct as \(f(x) = 0\) for infinite non-integer values of \(x\).

Key Concepts

Floor Function

The floor function, often denoted as \([x]\), is an important mathematical concept. It represents the greatest integer less than or equal to a given number \(x\). For instance, if \(x = 3.7\), then \([x] = 3\). Similarly, if \(x = -2.3\), \([x] = -3\). This function essentially 'rounds down' \(x\) to the nearest lower whole number.
Understanding the floor function is important in scenarios where you need to round numbers down, such as calculating ages (where a person's age is rounded down to the last complete year) or handling modules in computing.
When used in mathematical expressions, the floor function can affect the value of the expression significantly, requiring careful attention to detail when solving problems that incorporate it.

Greatest Integer Function

The greatest integer function is closely related to the floor function and is often called by the same name. It takes any real number \(x\) and gives back the greatest integer that is less than or equal to \(x\). For this function, the notation \([x]\) is frequently used. This means:

• If \(1.9\), then \([x] = 1\).
• If \(-4.2\), then \([x] = -5\).
• For integer values, \([x]\) is the same as \(x\). For example, \([3] = 3\).

These nuances are critical while evaluating expressions and equations using this function.
In our context, when solving \(f(x) = [x]^2 + [x+1] - 3 = 0\), we employ the greatest integer function which adjusts based on whether \(x\) is an integer or not, influencing the roots of the equation differently in both cases.

Quadratic Equation Solutions

Quadratic equations are fundamental in algebra and appear frequently in various mathematical considerations. A typical quadratic equation is of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
To find the solutions, known as roots, we utilize the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The term \(b^2 - 4ac\) is called the discriminant, which determines the nature of the roots. If it's positive, there are two distinct real roots; if zero, one real double root; and if negative, two complex roots.
In our exercise, converting \(x\) to \(n\) using the greatest integer function helps solve the quadratic equation \(n^2 + n - 2 = 0\). The roots \(n = 1\) and \(n = -2\) show that with non-integer \(x\), \(f(x)\) equals zero for infinitely many values, as explained in the solution.