Question

If $g(x) = x^{2} + x - 2$ and $\frac{1}{2}(gof)(x) = 2x^{2} - 5x + 2,$ then $f^{- 1}(x)$ is equal to

• A. $2x - 3$
• B. $2x + 3$
• C. $x,f(x) = 0$
• D. $\lbrack y\rbrack$

Answer 1

Answer: (A)

$2x - 3$

Answer 2

$g(x) = x^{2} + x - 2 \Rightarrow (gof)(x) = g\lbrack f(x)\rbrack = \lbrack f(x)\rbrack^{2} + f(x) - 2$

Given, $\frac{1}{2}(gof)(x) = 2x^{2} - 5x + 2$

$\therefore$ $\frac{1}{2}\lbrack f(x)\rbrack^{2} + \frac{1}{2}f(x) - 1 = 2x^{2} - 5x + 2$

$\Rightarrow \lbrack f(x)\rbrack^{2} + f(x) = 4x^{2} - 10x + 6$

$\Rightarrow$ $f(x)\lbrack f(x) + 1\rbrack = (2x - 3)\lbrack(2x - 3) + 1\rbrack \Rightarrow$ $f:R \rightarrow R$.