\(The function f is defined by f(x) = -3x2 + 2 for x ≤ -1.\)
\(The function g is defined by g(x) = -x2 - 1 for x ≤ -1.\)
\(Solve the equation fg(x) - gf(x) + 8 = 0.\)
Solution
First, calculate fg(x):
fg(x) = f(g(x)) = -3(-x2 - 1)2 + 2 = -3(x4 + 2x2 + 1) + 2 = -3x4 - 6x2 - 3 + 2 = -3x4 - 6x2 - 1.
Next, calculate gf(x):
gf(x) = g(f(x)) = -(-3x2 + 2)2 - 1 = -(9x4 - 12x2 + 4) - 1 = -9x4 + 12x2 - 5.
Substitute into the equation:
fg(x) - gf(x) + 8 = (-3x4 - 6x2 - 1) - (-9x4 + 12x2 - 5) + 8 = 6x4 - 18x2 + 12 = 0.
Factor the equation:
\(6(x4 - 3x2 + 2) = 0.\)
\(Let y = x2, then y2 - 3y + 2 = 0.\)
\(Factor the quadratic: (y - 1)(y - 2) = 0.\)
\(So, y = 1 or y = 2.\)
\(Thus, x2 = 1 or x2 = 2.\)
\(Therefore, x = -1 or x = -\sqrt{2}.\)
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