Question

If $ g(x) $ = $ x^2 + x-2 $ and $ \frac {1}{2}(gof) x = 2x^2-5x+2 $ then $f(x)$ is equal to

• A. 2x - 3
• B. 2x + 3
• C. 3x - 2
• D. 2x - 2

Answer 1

Answer: (A)

We have,
$g (x) =x^{2}+x-2$
and $\frac{1}{2}(go f) x=2x^{2}-5x+2$
$\Rightarrow \frac{1}{2}(g f(x))=2x^{2}-5x-2$
$\Rightarrow g (f(x))=4x^{2}-10x+4 \dots(i)$
and $g (f (x))=(f(x))^{2}+f (x)-2 \dots (ii)$
Eqs. (i) and (ii) are equal
$\therefore (f(x))^{2}+f (x)-2$
$=4x^{2}-10x+4$
$\Rightarrow (f(x))^{2}+f(x)=4x^{2}-10x+6$
$\Rightarrow \left\{f\left(x\right)\right\}^{2}+f\left(x\right)+\frac{1}{4}$
$=4x^{2}-10x+6+\frac{1}{4}$
$\Rightarrow \left(f\left(x\right)+\frac{1}{2}\right)^{2}=4x^{2}-10x+\frac{25}{4}$
$\Rightarrow \left(f\left(x\right)+\frac{1}{2}\right)^{2}=\left(2x-\frac{5}{2}\right)^{2}$
$\Rightarrow f \left(x\right)+\frac{1}{2}=2x-\frac{5}{2}$
$\Rightarrow f \left(x\right)=2x-3$