Algebra in 2018

Algebra Level 2

Given a function f : R → R f:\mathbb{R}\rightarrow \mathbb{R} . If f ( x ) + 2 f ( 2019 − x ) = x f\left( x\right) +2 f\left(2019-x\right)=x for every x x real, then what is the value of − 3 f ( 2018 ) -3 f\left(2018\right) ?

2019 No Correct Answer 2018 2016 2017

Erik Catos Lawijaya by Erik Catos Lawijaya , 19, Indonesia

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2 solutions

f ( x ) + 2 f ( 2019 − x ) = x f(x) + 2f(2019 - x) = x

Set x = 2018 , we have: f ( 2018 ) + 2 f ( 1 ) = 2018 f(2018) + 2f(1) = 2018

Set x = 1 , we have: f ( 1 ) + 2 f ( 2018 ) = 1 f(1) + 2f(2018) = 1

Use elimination method, we have: f ( 2018 ) = − 672 , f ( 1 ) = 1345 f(2018) = -672 , f(1) = 1345

So, the value of − 3 f ( 2018 ) -3f(2018) is 2016 \boxed{2016}

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James Wilson
Jan 3, 2021

We can substitute 2019 − x 2019-x in for x x in equation (1) to get equation (2).

f ( x ) + 2 f ( 2019 − x ) = x f(x)+2f(2019-x)=x (1)

f ( 2019 − x ) + 2 f ( 2019 − ( 2019 − x ) ) = 2019 − x f(2019-x)+2f(2019-(2019-x))=2019-x (2)

Equation (2) simplifies:

f ( 2019 − x ) + 2 f ( x ) = 2019 − x f(2019-x)+2f(x)=2019-x .

Multiply equation (2) by 2 and subtract it from equation (1) to obtain:

− 3 f ( x ) = 3 x − 4038 -3f(x)=3x-4038 .

Taking x = 2018 x=2018 gives the final answer.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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