f-x-1-f-x-1-x-2-find-f-1-x-

Question Number 117508 by bemath last updated on 12/Oct/20

f (x + 1) + f (x - 1) = x^{2}

find f^{- 1} (x) = ?

Answered by bobhans last updated on 12/Oct/20

let f (x) = {ax}^{2} + bx + c

f (x + 1) = a {(x + 1)}^{2} + b (x + 1) + c

= {ax}^{2} + (2 a + b) x + a + b + c

f (x - 1) = a {(x - 1)}^{2} + b (x - 1) + c

= {ax}^{2} + (b - 2 a) x + a - b + c

\Leftrightarrow f (x + 1) + f (x - 1) = {ax}^{2} + 2 bx + 2 a + 2 c

{\begin{cases} a = 1 \\ b = 0 \\ 2 a + 2 c = 0 \Rightarrow c = - 1 \end{cases}

f (x) = x^{2} - 1; x^{2} = y + 1

f^{- 1} (x) = \pm \sqrt{x + 1}; x ⩾ - 1

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