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Which of the following functions is NOT homogeneous?
- (A) x³ + 8x²y + 7y³
- (B) y² + x² + 5xy
- (C) xy/(x² + y²)
- (D) (2x - y + 1)/(2y - x + 1)
Correct answer: (D) (2x - y + 1)/(2y - x + 1)
Solution
A homogeneous function satisfies f(tx, ty) = tⁿ f(x, y). (A) is homogeneous of degree 3; (B) is homogeneous of degree 2; (C) numerator degree 2, denominator degree 2, so it is homogeneous of degree 0. (D) has constant terms (+1) in numerator and denominator, so replacing (x,y) with (tx,ty) does not produce a single power tⁿ times the original — the +1 does not scale. Hence (D) is not homogeneous.
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