Higher January 2022 Paper 2R Q23
23 The functions f and g are such that
\(\mathrm{f}(x) = x + 25 \qquad \mathrm{g}(x) = x^2 - 12x\)
The function h is such that \(\mathrm{h}(x) = \mathrm{fg}(x)\)
The domain of h is \(\{x : x \leqslant 6\}\)
Express the inverse function \(\mathrm{h}^{-1}\) in the form \(\mathrm{h}^{-1}(x) = \ldots\)
(4)
| Scheme | Marks |
|---|---|
| \(x^2 - 12x + 25\) | M1 |
| \((x - 6)^2 - 6^2\ (+ 25)\) or \((x - 6)^2 - 11\) or \(x^2 - 12x + (25 - y) = 0\) oe or \(y^2 - 12y + (25 - x) = 0\) oe | M1ft |
\((x - 6)^2 = y + 11\) or \((y - 6)^2 = x + 11\) or \(x = \dfrac{12 \pm \sqrt{144 - 4(25 - y)}}{2}\) oe or \(x = 6 \pm \sqrt{11 + y}\) | M1ft |
| \(6 - \sqrt{11 + x}\) | A1 |
| (4) | |
| (4 marks) |
Notes
M1: for substituting \(\mathrm{g}(x)\) into \(\mathrm{f}(x)\)
M1ft: (dep on M1) for a correct first step in order to complete the square. Allow \(y\) in place of \(x\).
or
Correctly setting up an equation = 0
M1ft: (dep on M2) for a correct rearrangement for their completed the square quadratic
or
correctly substituting into the quadratic formula (allow just + or just – instead of \(\pm\))
Allow same equations with \(x\) and \(y\) swapped
A1: oe must be in terms of \(x\) and have minus only before the square root.