for-x-R-satisfy-the-equation-f-x-3x-f-1-x-2-x-1-find-f-2019-

Question Number 83956 by jagoll last updated on 08/Mar/20

for x \in R satisfy the equation

f (x) + 3 x f (\frac{1}{x}) = 2 (x + 1)

find f (2019) .

Commented by mr W last updated on 08/Mar/20

f (x) + 3 x f (\frac{1}{x}) = 2 (x + 1) \dots (i)

f (\frac{1}{x}) + \frac{3}{x} f (x) = 2 (\frac{1 + x}{x}) \dots (i i)

(i) - 3 x (i i) :

\Rightarrow f (x) = \frac{1 + x}{2}

\Rightarrow f (2019) = \frac{1 + 2019}{2} = 1010

Answered by john santu last updated on 08/Mar/20

replace x by 2019 in eq (1)

f (2019) + 3 (2019) f (\frac{1}{2019}) = 2 (2020) (eq 2)

replace \frac{1}{x} by 2019 in eq (1)

f (\frac{1}{2019}) + \frac{3}{2019} f (2019) = 2 (\frac{2020}{2019})

now you are will get the value

f (2019)