CAT 2024 Slot 3 QA Question 1: For any non-zero real number $x$, let $f(x) + 2f(\dfrac{1}{x}) = 3x$. Then, the sum of all possible values of $x$ for which $f(x) = 3$, is

Question

For any non-zero real number $x$, let $f(x) + 2f(\dfrac{1}{x}) = 3x$. Then, the sum of all possible values of $x$ for which $f(x) = 3$, is

Accepted Answer

Replace $x$ by $\dfrac{1}{x}$ in the given relation to get a second equation in $f(x)$ and $f(\dfrac{1}{x})$.

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

Solving these two equations,

$f(x) = \dfrac{2}{x} - x$

Now set $f(x) = 3$.

$\dfrac{2}{x} - x = 3$
$2 - x^2 = 3x$

For any non-zero real number xxx, let f(x)+2f(1x)=3xf(x) + 2f(\dfrac{1}{x}) = 3xf(x)+2f(x1​)=3x. Then, the sum of all possible values of xxx for which f(x)=3f(x) = 3f(x)=3, is