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