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If $g(x)=x^2+x-2\:\:and\:\:\large\frac{1}{2}$$gof(x)=2x^2-5x+2$ then $ f(x)= ?$

(A) $2x-3$ (B) $2x+3$ (C) $2x^2+3x+1$ (D) $2x^2-3x-1$
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Given: $gof(x)=4x^2-10x+4$ and
$g(x)=x^2+x-2$
$\Rightarrow\:g(f)=f^2+f-2=4x^2-10x+4$
$\Rightarrow\:f^2+f-(4x^2-10x-6)=0$
Solving this quadratic equation on $f$ we get
$f(x)=\large\frac{-1\pm \sqrt{1+4(4x^2-10x-6)}}{2}$
$=\large\frac{-1\pm \sqrt{16x^2-40x+25}}{2}$
$=\large\frac{-1\pm \sqrt{4x-5)^2}}{2}$
$=\large\frac{-1\pm(4x-5)}{2}$$=2x-3\:\:or\:\:-2x+2$
answered May 28, 2013 by rvidyagovindarajan_1
 

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