| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Piecewise function inverses |
| Difficulty | Standard +0.3 This is a straightforward piecewise function question requiring students to find the range, sketch the inverse by reflection, and algebraically find inverse expressions for two simple functions (linear and reciprocal). While it involves multiple parts, each step uses standard techniques 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 existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(-5 \le f(x) \le 4\) For \(f(x)\) allow \(x\) or \(y\); allow \(<, [-5, 4], (-5,4)\) | B1 [1] | Allow less explicit answers (eg \(-5 \to 4\)) |
| (ii) \(f^{-1}(x)\) approximately correct (independent of f) Closed region between \((1, 1)\) and \((4, 4)\); line reaches \(x\)-axis | B1 DB1 [2] | Ignore line \(y = x\) |
| (iii) LINE: \(f^{-1}(x) = \frac{1}{3}(x + 2)\) | B1 | Allow \(y = \ldots\) but must be a function of \(x\) |
| for \(-5 \le x \le 1\) | B1B1 | cao but allow \(<\) |
| CURVE: \(5 - y = \frac{4}{x}\) OR \(x = 5 - \frac{4}{y}\) | M1 | |
| \(f^{-1}(x) = 5 - \frac{4}{x}\) oe | A1 [6] | cao |
| for \(1 < x \le 4\) | B1 | cao but allow \(< or <\) |
(i) $-5 \le f(x) \le 4$ For $f(x)$ allow $x$ or $y$; allow $<, [-5, 4], (-5,4)$ | B1 [1] | Allow less explicit answers (eg $-5 \to 4$)
(ii) $f^{-1}(x)$ approximately correct (independent of f) Closed region between $(1, 1)$ and $(4, 4)$; line reaches $x$-axis | B1 DB1 [2] | Ignore line $y = x$
(iii) LINE: $f^{-1}(x) = \frac{1}{3}(x + 2)$ | B1 | Allow $y = \ldots$ but must be a function of $x$
for $-5 \le x \le 1$ | B1B1 | cao but allow $<$
CURVE: $5 - y = \frac{4}{x}$ OR $x = 5 - \frac{4}{y}$ | M1 |
$f^{-1}(x) = 5 - \frac{4}{x}$ oe | A1 [6] | cao
for $1 < x \le 4$ | B1 | cao but allow $< or <$10\\
\includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-4_819_812_255_662}
The diagram shows the function f defined for $- 1 \leqslant x \leqslant 4$, where
$$f ( x ) = \begin{cases} 3 x - 2 & \text { for } - 1 \leqslant x \leqslant 1 \\ \frac { 4 } { 5 - x } & \text { for } 1 < x \leqslant 4 \end{cases}$$
(i) State the range of f .\\
(ii) Copy the diagram and on your copy sketch the graph of $y = \mathrm { f } ^ { - 1 } ( x )$.\\
(iii) Obtain expressions to define the function $\mathrm { f } ^ { - 1 }$, giving also the set of values for which each expression is valid.
\hfill \mbox{\textit{CAIE P1 2014 Q10 [9]}}