First slide

Functions (XII)

Question

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

Moderate

A

$2 x - 3$
B

$2 x + 3$
C

$2 x^{2} + 3 x + 1$
D

$2 x^{2} - 3 x - 1$

Solution

$g (x) = x^{2} + x - 2 a n d (g o f) (x) = 4 x^{2} - 10 x + 4$
$\Rightarrow g (f (x)) = {(f (x))}^{2} + f (x) - 2 = 4 x^{2} - 10 x + 4$
${(f (x))}^{2} + f (x) - (4 x^{2} - 10 x + 6) = 0 \Rightarrow$
$f (x) = \frac{- 1 \pm \sqrt{16 x^{2} - 40 x + 25}}{2} = \frac{- 1 \pm (4 x - 5)}{2} = 2 x - 3 o r - 2 x + 2$
$h e n c e f (x) = 2 x - 3$

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