If g ( x ) = 3 x 2 + 2 x - 3 , f ( 0 ) = - 3 a n d 4 g ( f ( x ) ) = 3 x 2 - 32 x + 72 , then f(g(2)) is equal to: [2026]

Question

If $g (x) = 3 x^{2} + 2 x - 3, f (0) = - 3 and 4 g (f (x)) = 3 x^{2} - 32 x + 72$ , then $f (g (2))$ is equal to: [2026]

Smart Achievers · Accepted Answer

(2)

$g (2) = 13$

$f (g (2)) = f (13)$

$Now 4 g (f (x)) = 3 x^{2} - 32 x + 72$

$4 [3 f^{2} (x) + 2 f (x) - 3] = 3 x^{2} - 32 x + 72$

$Let f (x) = t$

$12 t^{2} + 8 t - (3 x^{2} - 32 x + 84) = 0$

$f (x) = \frac{- 8 \pm \sqrt{64 + 48 (3 x^{2} - 32 x + 84)}}{24}$

$f (x) = \frac{- 8 \pm 4 (3 x - 16)}{24}$

$∵$ $f (0) = - 3$ $∴$ $we take +ve sign$

$∴$ $f (x) = \frac{- 8 + 4 (3 x - 16)}{24}$

$∴$ $f (13) = \frac{- 8 + 4 \cdot 23}{24} = \frac{84}{24} = \frac{7}{2}$

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