Question

Given that \( x \mathbin{@} y = x - y \), then find \( (x \mathbin{\$} y) + (x \mathbin{£} y) \).

• A. \( 2x^2 \)
• B. \( 2y^2 \)
• C. \( 2(x^2 + y^2) \)
• D. Cannot be determined

Hint:

If absolute values are involved, always consider multiple cases before concluding. If the answer changes with cases, the result cannot be determined.

Answer 1

The Correct Option is D

Step 1: Understand the operators
\[ x \mathbin{\$} y = x^2 + y^2,\quad x \mathbin{£} y = |x^2 - y^2| \] Step 2: Combine the expressions
\[ (x \mathbin{\$} y) + (x \mathbin{£} y) = x^2 + y^2 + |x^2 - y^2| \] Step 3: Analyze the cases
\text{If } x^2>y^2,\ \text{then } x^2 + y^2 + (x^2 - y^2) = 2x^2
\text{If } y^2>x^2,\ \text{then } x^2 + y^2 + (y^2 - x^2) = 2y^2 Hence, without knowing whether \( x^2>y^2 \) or vice versa, we cannot determine a unique value.