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Functional Equation

CAT 2008 Slot 1 · Quantitative Ability · Easy · Functions

This is an easy Quantitative Ability question from the CAT 2008 Slot 1 paper. It tests Functions. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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Let f(x)f(x) be a function satisfying f(x)f(y)=f(xy)f(x)f(y) = f(xy) for all real x,yx, y. If f(2)=4f(2) = 4, then what is the value of f(12)f\left(\frac{1}{2}\right)?

  1. A.

    0

  2. B.

    14\frac{1}{4}

  3. C.

    12\frac{1}{2}

  4. D.

    1

  5. E.

    cannot be determined

Answer

B

Explanation

Since f(x)f(y)=f(xy)f(x)f(y) = f(xy): For x=2,y=1x = 2, y = 1, f(2)f(1)=f(2)  ⟹  4f(1)=4  ⟹  f(1)=1f(2)f(1) = f(2) \implies 4f(1) = 4 \implies f(1) = 1.

For x=2,y=12x = 2, y = \frac{1}{2}: f(2)f(12)=f(2×12)=f(1)f(2)f\left(\frac{1}{2}\right) = f\left(2 \times \frac{1}{2}\right) = f(1) 4×f(12)=1  ⟹  f(12)=144 \times f\left(\frac{1}{2}\right) = 1 \implies f\left(\frac{1}{2}\right) = \frac{1}{4}.

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