The function f is defined by \(f(x) = 2x^2 - 16x + 23\) for \(x < 3\).
The function g is defined by \(g(x) = 2x + 4\) for \(x < -1\).
Find and simplify an expression for \(fg(x)\).
Solution
To find \(fg(x)\), we substitute \(g(x) = 2x + 4\) into \(f(x)\).
\(fg(x) = f(2x + 4) = 2((2x + 4)^2) - 16(2x + 4) + 23\).
First, calculate \((2x + 4)^2 = 4x^2 + 16x + 16\).
Then, \(2(4x^2 + 16x + 16) = 8x^2 + 32x + 32\).
Next, calculate \(-16(2x + 4) = -32x - 64\).
Combine these results: \(8x^2 + 32x + 32 - 32x - 64 + 23\).
Simplify: \(8x^2 + 32 - 64 + 23 = 8x^2 - 9\).
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