If-2f-x-f-1-x-3x-what-is-f-x-

Question Number 110436 by bemath last updated on 29/Aug/20

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

what is f (x)

Answered by john santu last updated on 29/Aug/20

r e p l a c e x b y \frac{1}{x}

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

\Leftrightarrow 4 f (x) + 2 f (\frac{1}{x}) = 6 x

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

________________-

\Rightarrow 3 f (x) = 6 x - \frac{3}{x}

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

Answered by mathmax by abdo last updated on 29/Aug/20

2 f (x) + f (\frac{1}{x}) = 3 x and 2 f (\frac{1}{x}) + f (x) = \frac{3}{x} (x \neq o) we get the systeme

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

Δ_{s} = | \begin{matrix} 2 1 \\ 1 2 \end{matrix} | = 3 \neq 0 Δ_{f (x)} = | \begin{matrix} 3 x 1 \\ \frac{3}{x} 2 \end{matrix} | = 6 x - \frac{3}{x}

Δ_{f (\frac{1}{x})} = | \begin{matrix} 2 3 x \\ 1 \frac{3}{x} \end{matrix} | = \frac{6}{x} - 3 x \Rightarrow f (x) = \frac{Δ_{f (x)}}{Δ} = \frac{6 x - \frac{3}{x}}{3}

f (x) = 2 x - \frac{1}{x}