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Definition of Functions

If fxy-1=f-...

Question

If $f (x y - 1) = f (- x) f (- y) + f (x) - y + 1$ and if $f (0) = 1$ , then $f (x) = ?$

A

x - 1

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B

- x + 1

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C

- x - 1

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D

x + 1

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E

1 - 2 x

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Solution

The correct option is A $- x + 1$
$f (x y - 1) = f (- x) f (- y) + f (x) - y + 1$
$f (y x - 1) = f (- y) f (- x) + f (y) - x + 1$
$\Rightarrow f (x) - y = f (y) - x$
$\Rightarrow$ Put $y = 0;$ $f (x) - 0 = f (o) - x$
$\Rightarrow f (x) = 1 - x$

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