| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a straightforward inverse function question requiring standard techniques: differentiation of a composite function using chain rule, interpreting the derivative's sign, and algebraic manipulation to find the inverse (cube root extraction). While it involves multiple parts, each step follows routine procedures taught in P1 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.07i Differentiate x^n: for rational n and sums1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(f'(x) = 9(3x + 2)^2\) or \(81x^2 + 108x + 36\) | B2, 1, 0 | for \(3(2x + 2)^2\), ×3 and \(\frac{d}{dx}(-5) = 0\) |
| Because of ( )² always +ve | B1√ | Allow for \(k(3x + 2)^2\). Tries numbers B0. |
| Therefore an increasing function. | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt[3]{y + 5} = 3x + 2\) | M1 | Attempt at making \(x\) the subject and completing to \(y = ...\) |
| \(f^{-1}(x) = \frac{\sqrt[3]{x + 5} - 2}{3}\) | A2, 1 | Loses one mark for each error. Leaving answer as \(f(y)\) is 1 error. |
| Domain of \(f^{-1}\) = range of \(f\) | B1 | co |
| \(\rightarrow x \geq -3\) | [4] |
$x \mapsto (3x + 2)^3 - 5$
**(i)** $f'(x) = 9(3x + 2)^2$ or $81x^2 + 108x + 36$ | B2, 1, 0 | for $3(2x + 2)^2$, ×3 and $\frac{d}{dx}(-5) = 0$
Because of ( )² always +ve | B1√ | Allow for $k(3x + 2)^2$. Tries numbers B0.
Therefore an increasing function. | [3]
**(ii)** $y = (3x + 2)^3 - 5$
$\sqrt[3]{y + 5} = 3x + 2$ | M1 | Attempt at making $x$ the subject and completing to $y = ...$
$f^{-1}(x) = \frac{\sqrt[3]{x + 5} - 2}{3}$ | A2, 1 | Loses one mark for each error. Leaving answer as $f(y)$ is 1 error.
Domain of $f^{-1}$ = range of $f$ | B1 | co
$\rightarrow x \geq -3$ | [4]6 The function f is such that $\mathrm { f } ( x ) = ( 3 x + 2 ) ^ { 3 } - 5$ for $x \geqslant 0$.\\
(i) Obtain an expression for $\mathrm { f } ^ { \prime } ( x )$ and hence explain why f is an increasing function.\\
(ii) Obtain an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE P1 2008 Q6 [7]}}