First slide

Functions (XII)

Question

If $f (x)$ is a polynomial satisfying $f (x) f (1 / x) = f (x) + f (1 / x) a n d f (3) = 28,$ then $f (4)$ is given by

Moderate

A

63
B

65
C

67
D

68

Solution

Any polynomial satisfying the functional equation $f (x) . f (1 / x) = f (x) + f (1 / x)$ is of the form $x^{n} + 1$ or $- x^{n} + 1$ If $28 = f (3) = - 3^{n} + 1$ then $3^{n} = - 27$ which is not possible for any n. Hence $28 = f (3) = 3^{n} + 1 \Rightarrow 3^{n} = 27 \Rightarrow n = 3 .$ Thus $f (x) = x^{3} + 1, s o f (4) = 4^{3} + 1 = 65$ .

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