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Nov 2023 p11 q9
689
The function f is defined by \(f(x) = 4x^2 - 12x + 13\) for \(p < x < q\), where \(p\) and \(q\) are constants. The function g is defined by \(g(x) = 3x + 1\) for \(x < 8\).
(b) Given that it is possible to form the composite function gf, find the least possible value of \(p\) and the greatest possible value of \(q\).
(c) Find an expression for \(gf(x)\).
Solution
(b) To find the least possible value of \(p\) and the greatest possible value of \(q\), we need to ensure that the range of \(g(x)\) fits within the domain of \(f(x)\). The function \(g(x) = 3x + 1\) is defined for \(x < 8\), so \(g(x) < 25\). We need \(p < g(x) < q\).
Solving \(4x^2 - 12x + 13 < 8\), we get:
\((2x - 3)^2 + 4 < 8\)
\((2x - 3)^2 < 4\)
\(-2 < 2x - 3 < 2\)
\(\frac{1}{2} < x < 2\frac{1}{2}\)
Thus, the least \(p = \frac{1}{2}\) and the greatest \(q = 2\frac{1}{2}\).
(c) To find \(gf(x)\), substitute \(g(x) = 3x + 1\) into \(f(x)\):
