4MA0 Higher June 2016 Paper 3R Q18
18 f is the function such that
\[\mathrm{f}(x) = \dfrac{x}{3x + 1}\](a) Find \(\mathrm{f}(0.5)\) (1)
(b) Find \(\mathrm{ff}(-1)\) (2)
(c) Find the value of \(x\) that cannot be included in any domain of f (1)
(d) Express the inverse function \(\mathrm{f}^{-1}\) in the form \(\mathrm{f}^{-1}(x) = \ldots\)
Show clear algebraic working. (3)
Show clear algebraic working. (3)
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{5}\) | B1oe |
| (1) |
| Scheme | Marks |
|---|---|
| M1 | |
| \(\dfrac{1}{5}\) | A1oe |
| (2) |
Notes
M1: \(\mathrm{f}(-1) = \dfrac{1}{2}\) or substitution of \(x = -1\) into \(\dfrac{\left(\dfrac{x}{3x + 1}\right)}{\left(\dfrac{3x}{3x + 1}\right) + 1}\)
| Scheme | Marks |
|---|---|
| \(-\dfrac{1}{3}\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
\(x = \dfrac{y}{3y + 1}\) \(x(3y + 1) = y\) or \(3xy + x = y\) or \(y = \dfrac{x}{3x + 1}\) \(y(3x + 1) = x\) or \(3xy + y = x\) | M1 |
| \(3xy - y = -x\) or \(y(3x - 1) = -x\) oe or \(3xy - x = -y\) or \(x(3y - 1) = -y\) oe | M1 |
| \(\dfrac{x}{1 - 3x}\) or \(\dfrac{-x}{3x - 1}\) | A1 |
| (3) |
Notes
M1: For writing function in the form \(x = \dfrac{y}{3y + 1}\) or \(y = \dfrac{x}{3x + 1}\) and multiplying both sides by the denominator
M1: For gathering terms in \(x\) or \(y\) (whichever is applicable) correctly
A1: Dep on M1
must be in terms of \(x\)