Higher January 2020 Paper 1 Q21
21 The functions f and g are such that
\(\mathrm{f}(x) = x^2 - 2x \qquad\qquad \mathrm{g}(x) = x + 3\)
The function h is such that \(\;\mathrm{h}(x) = \mathrm{fg}(x)\;\) for \(\;x \geqslant -2\)
Express the inverse function \(\;\mathrm{h}^{-1}(x)\;\) in the form \(\;\mathrm{h}^{-1}(x) = \ldots\)
(5)
| Scheme | Marks |
|---|---|
| \((\mathrm{fg}(x) =)\; (x + 3)^2 - 2(x + 3)\) oe | M1 |
| \((\mathrm{fg}(x) =)\; x^2 + 4x + 3\) | A1 |
| \((x + 2)^2 - 4 + 3\) or \((x + 2)^2 - 1\) or \(x^2 + 4x + (3 - y) = 0\) or \(y^2 + 4y + (3 - x) = 0\) | M1 |
\((x + 2)^2 = y + 1\) or \((y + 2)^2 = x + 1\) or \(x = \dfrac{-4 \pm \sqrt{16 - 4(3 - y)}}{2}\) or \(x = -2 \pm \sqrt{1 + y}\) | M1 |
| \(-2 + \sqrt{x + 1}\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for substituting \(\mathrm{g}(x)\) into \(\mathrm{f}(x)\)
A1: Allow \(y^2 + 4y + 3\)
M1: ft (dep on M1) for correctly completing the square on their 3 term quadratic
or
Correctly setting up an equation
M1: ft (dep on M2) for a correct rearrangement for their completed the square quadratic
or
correctly substituting into the quadratic formula
Allow same equations with \(x\) and \(y\) swapped
A1: oe