Q is a rational function with x &#x22C5;<!-- ⋅ --> Q ( x + 2018 ) = (

Eliaszowyr1

Eliaszowyr1

Answered question

2022-05-24

Q is a rational function with x Q ( x + 2018 ) = ( x 2018 ) Q ( x ) , x 2018  and  0. If Q ( 1 ) = 1. what is Q(2017)?
I tried substituting values in for Q(x) but there is no way to have both Q(1) and Q(2017) in the same equation, perhaps im missing a property of a rational function since i haven't used it yet.

Answer & Explanation

concludirgt

concludirgt

Beginner2022-05-25Added 11 answers

If F ( x ) = x Q ( x + 2018 ), then F ( x ) = F ( x 2018 ). Hence F is a periodic rational function and therefore F is constant. If F ( x ) = c, then we have:
x Q ( x + 2018 ) = c Q ( x + 2018 ) = c x Q ( x ) = c x 2018
From Q ( 1 ) = 1 we get c = 2017, hence Q ( x ) = 2017 / ( 2018 x ); in particular, Q ( 2017 ) = 2017.

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