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June 2016 p12 q11
727
The function g is defined by \(g : x \mapsto 6x - x^2 - 5\) for \(x \geq k\), where \(k\) is a constant.
(iii) Express \(6x - x^2 - 5\) in the form \(a - (x - b)^2\), where \(a\) and \(b\) are constants.
(iv) State the smallest value of \(k\) for which \(g\) has an inverse.
(v) For this value of \(k\), find an expression for \(g^{-1}(x)\).
Solution
(iii) To express \(6x - x^2 - 5\) in the form \(a - (x - b)^2\), we complete the square:
\(6x - x^2 - 5 = -(x^2 - 6x) - 5\)
\(= -(x^2 - 6x + 9 - 9) - 5\)
\(= -(x - 3)^2 + 9 - 5\)
\(= 4 - (x - 3)^2\)
Thus, \(a = 4\) and \(b = 3\).
(iv) The function \(g\) has an inverse when it is one-to-one. The vertex of the parabola \(y = 6x - x^2 - 5\) is at \(x = 3\). Therefore, the smallest value of \(k\) for which \(g\) is one-to-one is \(k = 3\).
(v) For \(k = 3\), the function \(g(x) = 4 - (x - 3)^2\). To find the inverse, set \(y = 4 - (x - 3)^2\) and solve for \(x\):
