| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.8 This is a straightforward inverse function question requiring routine differentiation to show the function is monotonic (one-to-one), then algebraic manipulation to find the inverse. Finding domain/range of the inverse requires evaluating endpoints but involves no conceptual difficulty. Standard P1/AS-level material with clear procedural steps. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.07l Derivative of ln(x): and related functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(x) = \frac{15}{2x+3}\) | ||
| (i) \(f'(x) = \frac{-15}{(2x+3)^2} \times 2\) | B1 B1 | Without the "\(\times 2\)". For "\(\times 2\)" (indep of 1st B1). |
| \(( )^2\) always \(+\)ve \(\rightarrow f'(x) < 0\) (No turning points) – therefore an inverse | B1✓ | ✓ providing \(( )^2\) in \(f'(x)\), 1–1 insufff. |
| [3] | ||
| (ii) \(y = \frac{15}{2x+3} \rightarrow 2x + 3 = \frac{15}{y}\) | M1 | Order of ops – allow sign error |
| \(\rightarrow x = \frac{15 - 3y}{2} = \frac{15-3x}{2x}\) | A1 | co as function of \(x\). Allow \(y = \ldots\) |
| (Range) \(0 \leq f^{-1}(x) \leq 6\). Allow \(0 \leq y \leq 6, [0,6]\) | B1 | For range/domain ignore letters unless range/domain not identified |
| (Domain) \(1 \leq x \leq 5\). Allow \([1, 5]\) | B1 | |
| [4] |
$f(x) = \frac{15}{2x+3}$ | | |
(i) $f'(x) = \frac{-15}{(2x+3)^2} \times 2$ | B1 B1 | Without the "$\times 2$". For "$\times 2$" (indep of 1st B1). |
$( )^2$ always $+$ve $\rightarrow f'(x) < 0$ (No turning points) – therefore an inverse | B1✓ | ✓ providing $( )^2$ in $f'(x)$, 1–1 insufff. |
| | [3] |
(ii) $y = \frac{15}{2x+3} \rightarrow 2x + 3 = \frac{15}{y}$ | M1 | Order of ops – allow sign error |
$\rightarrow x = \frac{15 - 3y}{2} = \frac{15-3x}{2x}$ | A1 | co as function of $x$. Allow $y = \ldots$ |
(Range) $0 \leq f^{-1}(x) \leq 6$. Allow $0 \leq y \leq 6, [0,6]$ | B1 | For range/domain ignore letters unless range/domain not identified |
(Domain) $1 \leq x \leq 5$. Allow $[1, 5]$ | B1 | |
| | [4] |5 A function f is such that $\mathrm { f } ( x ) = \frac { 15 } { 2 x + 3 }$ for $0 \leqslant x \leqslant 6$.\\
(i) Find an expression for $\mathrm { f } ^ { \prime } ( x )$ and use your result to explain why f has an inverse.\\
(ii) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$, and state the domain and range of $\mathrm { f } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE P1 2014 Q5 [7]}}