If-g-x-x-2-x-2-and-1-2-gof-x-2x-2-5x-2-then-prove-that-f-x-2x-3-

Question Number 13598 by Tinkutara last updated on 21/May/17

If g (x) = x^{2} + x - 2 and

\frac{1}{2} g o f (x) = 2 x^{2} - 5 x + 2, then prove

that f (x) = 2 x - 3 .

Commented by mrW1 last updated on 21/May/17

\frac{1}{2} g o f = \frac{1}{2} (f^{2} + f - 2) = 2 x^{2} - 5 x + 2

f^{2} + f - 2 (2 x^{2} - 5 x + 3) = 0

f = \frac{- 1 \pm \sqrt{1 + 4 \times 2 (2 x^{2} - 5 x + 3)}}{2}

f = \frac{- 1 \pm \sqrt{16 x^{2} - 40 x + 25}}{2}

f (x) = \frac{- 1 \pm (4 x - 5)}{2} = \frac{4 x - 6}{2} = 2 x - 3 o r \frac{- 4 x + 4}{2} = 2 (1 - x)

t h a t m e a n s t h e r e a r e 2 s o l u t i o n s :

f (x) = 2 x - 3

o r f (x) = 2 (1 - x)

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