MCQ
If $g(x)=x^2+x-2$ and $\frac{1}{2} \operatorname{gof}(x)=2 x^2-5 x+2$, then $f(x)$ is equal to:
- ✓$2 x-3$
- B$2x + 3$
- C$2 x^2+3 x+1$
- D$2 x^2-3 x-1$
✓
Answer
Correct option: A.
$2 x-3$
We will solve this problem by the trial$-$and$-$error method.
Let us check option $(a)$ first.
If $f(x) = 2x - 3$
$\frac{1}{2}(\text{gof})(x)=\text{g(f(x))}$
$=\frac{1}{2}\text{g}(2\text{x}-3)$
$=\frac{1}{2}\big[(2\text{x}-3)^2+(2\text{x}-3)-2\big]$
$=\frac{1}{2}[4\text{x}^2+9-12\text{x}+2\text{x}-3-2]$
$=\frac{1}{2}[4\text{x}^2-10\text{x}+4]$
$=2\text{x}^2-5\text{x}+2$
The given condition is satisfied by $(a).$
Let us check option $(a)$ first.
If $f(x) = 2x - 3$
$\frac{1}{2}(\text{gof})(x)=\text{g(f(x))}$
$=\frac{1}{2}\text{g}(2\text{x}-3)$
$=\frac{1}{2}\big[(2\text{x}-3)^2+(2\text{x}-3)-2\big]$
$=\frac{1}{2}[4\text{x}^2+9-12\text{x}+2\text{x}-3-2]$
$=\frac{1}{2}[4\text{x}^2-10\text{x}+4]$
$=2\text{x}^2-5\text{x}+2$
The given condition is satisfied by $(a).$
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