Functional equation evaluation

CAT 2020 Slot 3 · QA · Medium · Functions

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If f(x+y)=f(x)f(y)f(x+y) = f(x)f(y) and f(5)=4f(5) = 4, then f(10)−f(−10)f(10) - f(-10) is equal to

  1. A.

    3

  2. B.

    0

  3. C.

    14.0625

  4. D.

    15.9375

Answer

D

Explanation

The functional equation f(x+y)=f(x)f(y)f(x+y) = f(x)f(y) implies f(x)=axf(x) = a^x. f(5)=a5=4f(5) = a^5 = 4. f(10)=f(5+5)=f(5)2=42=16f(10) = f(5+5) = f(5)^2 = 4^2 = 16. f(0)=f(5−5)=f(5)f(−5)=1  ⟹  f(−5)=1/4f(0) = f(5-5) = f(5)f(-5) = 1 \implies f(-5) = 1/4. f(−10)=f(−5)2=(1/4)2=1/16=0.0625f(-10) = f(-5)^2 = (1/4)^2 = 1/16 = 0.0625.

f(10)−f(−10)=16−0.0625=15.9375f(10) - f(-10) = 16 - 0.0625 = 15.9375.

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