Find-the-function-f-x-if-3f-x-1-f-1-x-x-2x-

Question Number 130894 by EDWIN88 last updated on 30/Jan/21

F i n d t h e f u n c t i o n f (x) i f

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

Commented by benjo_mathlover last updated on 30/Jan/21

այս հարցը շատ հետաքրքիր է

Commented by EDWIN88 last updated on 30/Jan/21

շնորհակալություն

Answered by mr W last updated on 30/Jan/21

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

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

3 \times (i) + (i i) :

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

f (x - 1) = \frac{1}{4} (3 x + \frac{1}{x})

\Rightarrow f (x) = \frac{1}{4} (3 (x + 1) + \frac{1}{x + 1})

Commented by EDWIN88 last updated on 30/Jan/21

��nice

Answered by benjo_mathlover last updated on 30/Jan/21

replace \frac{1 - x}{x} by t \Rightarrow x = \frac{1}{t + 1}

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

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

replace x - 1 by t \Rightarrow x = t + 1

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

3 f (x) - f (\frac{- t}{t + 1}) = 2 t + 2 \dots (ii)

\Leftrightarrow 3 \times (ii) \Rightarrow 9 f (t) - 3 f (\frac{- t}{t + 1}) = 6 t + 6

\Leftrightarrow 1 \times (i) \Rightarrow - f (t) + 3 f (\frac{- t}{t + 1}) = \frac{2}{t + 1}

__________________________+

\Leftrightarrow 8 f (t) = 6 t + 6 + \frac{2}{t + 1}

\Leftrightarrow f (x) = \frac{3 x + 3}{4} + \frac{1}{4 x + 4} = \frac{12 {(x + 1)}^{2} + 4}{16 (x + 1)}

= \frac{{3 x}^{2} + 6 x + 4}{4 x + 4}