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Nov 2011 p12 q2
696
The functions f and g are defined for \(x \in \mathbb{R}\) by
\(f : x \mapsto 3x + a,\)
\(g : x \mapsto b - 2x,\)
where \(a\) and \(b\) are constants. Given that \(ff(2) = 10\) and \(g^{-1}(2) = 3\), find
the values of \(a\) and \(b\),
an expression for \(fg(x)\).
Solution
(i) To find \(a\), we use \(ff(2) = 10\). First, calculate \(f(2) = 3(2) + a = 6 + a\). Then, \(ff(2) = f(f(2)) = f(6 + a) = 3(6 + a) + a = 18 + 3a + a = 18 + 4a\). Given \(ff(2) = 10\), we have:
\(18 + 4a = 10\)
\(4a = 10 - 18\)
\(4a = -8\)
\(a = -2\)
To find \(b\), use \(g^{-1}(2) = 3\). The inverse function \(g^{-1}(x)\) is found by solving \(y = b - 2x\) for \(x\):
\(x = \frac{b - y}{2}\)
Given \(g^{-1}(2) = 3\), we have:
\(3 = \frac{b - 2}{2}\)
\(6 = b - 2\)
\(b = 8\)
(ii) To find \(fg(x)\), substitute \(g(x) = b - 2x\) into \(f(x)\):
