Question:
If $g(x)=x^{2}+x-2$ and $\frac{1}{2} g o f(x)=2 x^{2}-5 x+2$, then $f(x)$ is equal to
(a) $2 x-3$
(b) $2 x+3$
(c) $2 x^{2}+3 x+1$
(d) $2 x^{2}-3 x-1$
Solution:
We will solve this problem by the trial-and-error method.
Let us check option (a) first.
If $f(x)=2 x-3$
$\frac{1}{2}(g o f)(x)=g(f(x))$
$=\frac{1}{2} g(2 x-3)$
$=\frac{1}{2}\left[(2 x-3)^{2}+(2 x-3)-2\right]$
$=\frac{1}{2}\left[4 x^{2}+9-12 x+2 x-3-2\right]$
$=\frac{1}{2}\left[4 x^{2}-10 x+4\right]$
$=2 x^{2}-5 x+2$
The given condition is satisfied by (a).
So, the answer is (a).
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