Question

If $g\left(x\right)=x^2+x-2$ and $\frac{1}{2}(g\circ f)\left(x\right)=2x^2-5x+2$, then $f\left(x\right)$ is

• A. $2x-3$
• B. $2x+3$
• C. $2x^2+3x+1$
• D. $2x^2+3x-1$

Answer 1

The correct option is A $2x-3$

$\frac{1}{2}(g\circ f)\left(x\right)=2x^2-5x+2$

$\Rightarrow \frac{1}{2}g\left[f\right(x\left)\right]=2x^2-5x+2$

$[{\left\{f\right(x\left)\right\}}^2+\left\{f\right(x\left)\right\}-2]=2[2x^2-5x+2][\because g(x)=x^2+x-2]$

$\Rightarrow f{\left(x\right)}^2+f\left(x\right)-(4x^2-10x+6)=0$ ....represents quadratic equation

We have a formula for solving quadratic equation $a{x}^2+bx+c=0$ is

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\therefore f\left(x\right)=\frac{-1\pm \sqrt{1+4(4x^2-10x+6)}}{2}$

$=\frac{-1\pm (4x-5)}{2}=2x-3$ or $2-2x$