4MA0 Higher June 2016 Paper 4 Q17
17 f is the function such that \(\mathrm{f}(x) = 2x - 5\)
g is the function such that \(\mathrm{g}(x) = x^2 - 10\)
(a) Find \(\mathrm{f}(4)\) (1)
(b) Find \(\mathrm{fg}(-4)\) (2)
(c) Express the inverse function \(\mathrm{f}^{-1}\) in the form \(\mathrm{f}^{-1}(x) = \ldots\) (2)
(d) Solve \(\mathrm{gf}(x) = -1\) (4)
| Scheme | Marks |
|---|---|
| 3 | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| M1 | |
| 7 | A1 |
| (2) |
Notes
M1: For \(2((-4)^2 - 10) - 5\) oe or
\((-4)^2 - 10\) or 6
| Scheme | Marks |
|---|---|
\(2x = y + 5\) or \(2y = x + 5\) or \(\dfrac{1}{2}(y + 5)\) | M1 |
| \(\dfrac{1}{2}(x + 5)\) | A1 |
| (2) |
Notes
A1: oe
| Scheme | Marks |
|---|---|
| \((2x - 5)^2 - 10\) (= −1) or \(4x^2 - 10x - 10x + 25 - 10\) (= −1) | M1 |
| \(4x^2 - 20x + 16\) (= 0) or \(2x^2 - 10x + 8\) (= 0) or \(x^2 - 5x + 4\) (= 0) or \((2x - 5)^2 = 9\) | M1 |
\((4x - 4)(x - 4)\) (= 0) or \((2x - 2)(x - 4)\) (= 0) or \((x - 4)(x - 1)\) (= 0) or \(2x - 5 = \pm 3\) \(\dfrac{--5 \pm \sqrt{(-5)^2 - 4(1)(4)}}{2(1)}\) (may be partially evaluated; condone lack of brackets around negative numbers) | M1 |
| \(x = 1\), \(x = 4\) | A1 |
| (4) |
Notes
M1: For a correct expression for \(\mathrm{gf}(x)\)
M1: For a correct 3 part quadratic or
For \((2x - 5)^2 = 9\)
M1: For factorising a correct equation or for use of quadratic formula with a correct equation or
For \(2x - 5 = \pm 3\)
Alternative method
| Scheme | Marks |
|---|---|
| Eg \(a^2 - 10 = -1\) oe | M1 |
| \(a^2 = 9\) | M1 |
| \(2x - 5 = \pm 3\) | M1 |
| \(x = 1\), \(x = 4\) | A1 |
Notes
M1: For a correct equation relating to \(\mathrm{g}(a) = -1\)
M1: For \(a^2 = 9\)
M1: For \(2x - 5 = \pm 3\)