Comprehension
Let x and y be real numbers. Let f(x,y) = |x+y|, F(f(x,y)) = f(x,y) and G(f(x,y)) = −F(f(x,y)).
Question: 1
Which of the following is true?
Which of the following is true?
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Pay attention to the properties of functions and their interactions when solving functional equations, especially with multiplication and symmetry.
Updated On: Aug 5, 2025
- \( F(f(x, y)) \cdot G(f(x, y)) = -F(f(x, y)) \cdot G(f(x, y)) \)
- \( F(f(x, y)) \cdot G(f(x, y)) \geq -F(f(x, y)) \cdot G(f(x, y)) \)
- \( F(f(x, y)) \cdot G(f(x, y)) \neq G(f(x, y)) \cdot F(f(x, y)) \)
- \( F(f(x, y)) + G(f(x, y)) + f(x, y) = f(x, -y) \)
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The Correct Option is A
Solution and Explanation
This problem involves the properties of functions \( f(x, y) \) and \( G(f(x, y)) \). By substituting in the expressions for these functions, we arrive at the conclusion that \( F(f(x, y)) \cdot G(f(x, y)) = -F(f(x, y)) \cdot G(f(x, y)) \), making option (1) correct.
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Question: 2
What is the value of: \( f(G(f(1, 0))), f(F(1, 2)), G(f(1, 2))) \)?
What is the value of: \( f(G(f(1, 0))), f(F(1, 2)), G(f(1, 2))) \)?
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When working with composed functions, evaluate from the innermost function outwards to simplify the expression step by step.
Updated On: Aug 5, 2025
- 3
- 2
- 1
- 0
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The Correct Option is C
Solution and Explanation
To calculate the given expression, we substitute values into the functions \( f \) and \( G \). After evaluating each function step by step, we find that the value of the expression is 1.
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Question: 3
Which of the following has \( x^2 \) as the result?
Which of the following has \( x^2 \) as the result?
Show Hint
When solving for expressions with unknown functions, carefully simplify each term and check if the final expression matches the desired result.
Updated On: Aug 5, 2025
- \( F(f(x, -x)), G(f(x, -x)) \)
- \( F(f(x, x)), G(f(x, x)) \cdot 4 \)
- \( F(f(x, x)) \cdot G(f(x, x)) \)
- \( f(x), x, f(x) \)
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The Correct Option is C
Solution and Explanation
We evaluate each option based on the functions provided. After simplifying the expressions, we find that option (3) yields \( x^2 \) as the result.
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