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Algebra Level 3

Let f f be a real-valued function such that

f ( x − f ( y ) ) = f ( x ) − x f ( y ) \large f(x-f(y)) = f(x) - x f(y)

for any real numbers x x and y y . If f ( 0 ) = 3 f(0) = 3 , determine f ( 2017 ) − f ( 2014 ) f(2017) - f(2014)


The answer is 6051.

Benedict Dimacutac by Benedict Dimacutac , 21, Philippines

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1 solution

Chan Tin Ping
Nov 4, 2017

Put y=0, we get f ( x − f ( 0 ) ) = f ( x ) − x f ( 0 ) f ( x − 3 ) = f ( x ) − 3 x \begin{aligned} f(x-f(0))&=f(x)-xf(0)\\ f(x-3)&=f(x)-3x \end{aligned}

Now evaluate f(2017)-f(2014). f ( 2017 ) − f ( 2014 ) = f ( 2017 ) − f ( 2017 − 3 ) = f ( 2017 ) − f ( 2017 ) + 3 × 2017 = 6051 \begin{aligned} &f(2017)-f(2014)\\ =&f(2017)-f(2017-3)\\ =&f(2017)-f(2017)+3\times 2017\\ =&6051 \end{aligned}

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