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Given \( f(x) = 3x^2 - 2x + 1 \) and \( g(x) = x - 4 \), calculate \( g(f(3)) \).
\[ f(3) = 3(3)^2 - 2(3) + 1 = 3 \times 9 - 6 + 1 = 27 - 6 + 1 = 22. \]
Next, substitute \( f(3) \) into \( g(x) \).
\[ g(f(3)) = g(22) = 22 - 4 = 18. \]
Thus, \( g(f(3)) = \boxed{18} \).
Determine the largest integer \( x \) that satisfies the inequality \( 2x^2 - 9x + 7 < 0 \).
Solve the quadratic inequality \( 2x^2 - 9x + 7 < 0 \).
First, find the roots of the equation \( 2x^2 - 9x + 7 = 0 \) using the quadratic formula:
where \( a = 2 \), \( b = -9 \), \( c = 7 \).
\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4 \times 2 \times 7}}{2 \times 2} = \frac{9 \pm \sqrt{81 - 56}}{4} = \frac{9 \pm \sqrt{25}}{4} = \frac{9 \pm 5}{4}. \]