| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| Session | November |
| Marks | 8 |
| 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 to verify monotonicity, then algebraic manipulation to find the inverse and determine its domain from the original range. While it involves multiple steps, each is routine for P1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(f'(x) = 3(2x-3)^2 \times 2\) | B1B1 | B1 for answer without ×2. B1 for ×2. |
| Answer | Marks | Guidance |
|---|---|---|
| Therefore an increasing function. (or t.p. at (1.5, -8) M1. Compares with \(y\) values at 2, or \(4 +\) conclusion A1) | B1 B1 | Realising 'increasing' \(\rightarrow\) +ve gradient) Stating \(()^2\) +ve for all \(x\). All complete. |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x - 3 = \sqrt[3]{y+8}\) | M1 DM1 | Attempt to make \(x\) subject. Order of operations correct. |
| \(\rightarrow f^{-1}(x) = \frac{\sqrt[3]{x+8}+3}{2}\) | A1 | Co – needs \(x\) not \(y\). |
| Domain -7 ≤ x ≤ 117 | B1 |
$f: x \mapsto (2x-3)^3 - 8$
(i) $f'(x) = 3(2x-3)^2 \times 2$ | B1B1 | B1 for answer without ×2. B1 for ×2.
Since $()^2$ is +ve, $f(x)$ +ve for all $x$
Therefore an increasing function. (or t.p. at (1.5, -8) M1. Compares with $y$ values at 2, or $4 +$ conclusion A1) | B1 B1 | Realising 'increasing' $\rightarrow$ +ve gradient) Stating $()^2$ +ve for all $x$. All complete. | [4]
(ii) $y = (2x-3)^3 - 8,$
$2x - 3 = \sqrt[3]{y+8}$ | M1 DM1 | Attempt to make $x$ subject. Order of operations correct.
$\rightarrow f^{-1}(x) = \frac{\sqrt[3]{x+8}+3}{2}$ | A1 | Co – needs $x$ not $y$. | [4]
Domain -7 ≤ x ≤ 117 | B1 |8 A function f is defined by $\mathrm { f } : x \mapsto ( 2 x - 3 ) ^ { 3 } - 8$, for $2 \leqslant x \leqslant 4$.\\
(i) Find an expression, in terms of $x$, for $\mathrm { f } ^ { \prime } ( x )$ and show that f is an increasing function.\\
(ii) Find an expression, in terms of $x$, for $\mathrm { f } ^ { - 1 } ( x )$ and find the domain of $\mathrm { f } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE P1 2005 Q8 [8]}}