Question

If $g\left(x\right)=x^2+x-2\mathrm{and}\frac{1}{2}gof\left(x\right)=2x^2-5x+2$, then f(x) is equal to
(a) $2x-3$
(b) $2x+3$
(c) $2x^2+3x+1$
(d) 2 $x^2-3x-1$

Answer 1

We will solve this problem by the trial-and-error method.
Let us check option (a) first.
$$\begin{gathered} \text{If}f\left(x\right)=2x-3 \\ \frac{1}{2}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right) \\ =\frac{1}{2}g\left(2x-3\right) \\ =\frac{1}{2}\left[{\left(2x-3\right)}^2+\left(2x-3\right)-2\right] \\ =\frac{1}{2}\left[4x^2+9-12x+2x-3-2\right] \\ =\frac{1}{2}\left[4x^2-10x+4\right] \\ =2x^2-5x+2 \end{gathered}$$
The given condition is satisfied by (a).
So, the answer is (a).