Functions f and g are defined for \(x \in \mathbb{R}\) by
\(f : x \mapsto 2x + 3\),
\(g : x \mapsto x^2 - 2x\).
Express \(gf(x)\) in the form \(a(x + b)^2 + c\), where \(a, b\) and \(c\) are constants.
Solution
First, substitute \(f(x) = 2x + 3\) into \(g(x) = x^2 - 2x\):
\(gf(x) = (2x + 3)^2 - 2(2x + 3)\).
Expand \((2x + 3)^2\):
\((2x + 3)^2 = 4x^2 + 12x + 9\).
Expand \(-2(2x + 3)\):
\(-2(2x + 3) = -4x - 6\).
Combine the expressions:
\(gf(x) = 4x^2 + 12x + 9 - 4x - 6\).
Simplify:
\(gf(x) = 4x^2 + 8x + 3\).
Complete the square for \(4x^2 + 8x + 3\):
Factor out 4 from the quadratic terms:
\(4(x^2 + 2x) + 3\).
Complete the square inside the parentheses:
\(x^2 + 2x = (x + 1)^2 - 1\).
Substitute back:
\(4((x + 1)^2 - 1) + 3\).
Expand and simplify:
\(4(x + 1)^2 - 4 + 3 = 4(x + 1)^2 - 1\).
Thus, \(gf(x) = 4(x + 1)^2 - 1\).
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