Problem 36 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f\left(2 x+\frac{y}{8}, 2 x-\frac{y}{8}\right)=x y\), then \(f(m, n)+f(n, m)=0\) (A) only when \(m=n\) (B) only when \(m \neq n\) (C) only when \(m=-n\) (D) for all \(m\) and \(n\).

Step-by-Step Solution

Answer

The condition holds for all \( m \) and \( n \), so the answer is (D).

Step 1: Understand the Problem Statement

We are given a function in terms of transformed variables. The goal is to find under what conditions the equation \( f(m, n) + f(n, m) = 0 \) holds true.

Step 2: Identify Function Transformation

The function \( f \) is given as \( f(2x + \frac{y}{8}, 2x - \frac{y}{8}) = xy \). We need to analyze how the original function can transform different variables \( m \) and \( n \).

Step 3: Express m and n in Terms of x and y

To find \( f(m, n) \), let's assume \( m = 2x + \frac{y}{8} \) and \( n = 2x - \frac{y}{8} \). Then, \( f(m, n) = xy \) becomes important in further analysis.

Step 4: Calculate Function Values

From the transformation, \( f(m, n) = xy \) leads us to find expressions for \( x \) and \( y \) in terms of \( m \) and \( n \). Thus, solve: \( 2x + \frac{y}{8} = m \) and \( 2x - \frac{y}{8} = n \).

Step 5: Solve System of Equations

Adding the equations gives \( 4x = m + n \) or \( x = \frac{m+n}{4} \). Subtracting them gives \( y = 8(m - n) \).

Step 6: Substitute Back into the Function

Substitute these expressions back into \( xy \) which means: \( xy = \left(\frac{m+n}{4}\right) \times 8(m-n) = 2(m+n)(m-n) \).

Step 7: Calculate f(n, m) and Analyze the Result

Since the setup is symmetric for \( f(n, m) \, \) use the same steps for terms swapped: \( f(n, m) = 2(n+m)(n-m) = -2(m+n)(m-n) \).

Step 8: Analyze the Sum f(m, n) + f(n, m)

Now, compute \( f(m, n) + f(n, m) = 2(m+n)(m-n) + (-2(m+n)(m-n)) = 0 \). It confirms the equation is always true.

Key Concepts

Function Transformation

In algebra, a function transformation alters a function's formula to produce a new graph. This concept helps us understand how inputs change and transform under the given rules. In our problem, the function transformation can be seen in the form of variables replaced inside the function. The function is given as \( f(2x + \frac{y}{8}, 2x - \frac{y}{8}) = xy \).This transformation plays a key role in solving the equation \( f(m, n) + f(n, m) = 0 \). Here, \( m \) and \( n \) are the transformed variables in terms of \( x \) and \( y \). The trick is to express the original function's input as these new variables \( m \) and \( n \), which is achieved by setting \( m = 2x + \frac{y}{8} \) and \( n = 2x - \frac{y}{8} \). By doing this transformation, we simplify the calculations necessary to solve the problem at hand.

System of Equations

Systems of equations are sets of equations with multiple variables that we solve simultaneously. They are used to find specific values of variables that satisfy all equations in the system. In our exercise, once we've expressed \( m \) and \( n \) in terms of \( x \) and \( y \), these transformations naturally lead us to a system of equations. To solve for \( x \) and \( y \), we consider the equations:

• \( 2x + \frac{y}{8} = m \)
• \( 2x - \frac{y}{8} = n \)

Adding and subtracting these equations enables us to isolate \( x \) and \( y \):

• Adding them gives \( 4x = m + n \), so \( x = \frac{m+n}{4} \)
• Subtracting them gives \( y = 8(m - n) \)

This process allows us to substitute back into the function and look for patterns or conditions that apply to the overall equation that we're investigating.

Symmetry in Functions

Symmetry in functions is a powerful concept that can simplify computation and analysis. In this context, symmetry indicates whether flipping inputs doesn't change the function's outcome. When analyzing \( f(m, n) + f(n, m) = 0 \), identifying symmetrical properties saves time by predicting results without calculation.Here, despite switching \( m \) and \( n \), we use the same functional form for both permutations, indicating symmetry affects results directly. Therefore:

• Calculating \( f(m, n) \) results in \( 2(m+n)(m-n) \)
• Similarly, \( f(n, m) \) transforms to \(-2(m+n)(m-n)\)

This transformation shows that swapping the inputs gives the opposite function value ensuring that their summation remains zero, regardless of the specific values originally assigned to \( m \) or \( n \).