| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Sketch function and inverse graphs |
| Difficulty | Moderate -0.8 This is a straightforward question on linear functions requiring routine algebraic manipulation to find the inverse, solving a simple simultaneous equation for intersection, and sketching two linear graphs with reflection in y=x. All techniques are standard with no problem-solving insight needed, making it easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{4-x}{5}\) | B1 | OE |
| Equate a valid attempt at \(f^{-1}\) with \(f\), or with \(x\), or \(f\) with \(x\) → \(\left(\frac{2}{3}, \frac{2}{3}\right)\) or \((0.667, 0.667)\) | M1, A1 | Equating and an attempt to solve as far as \(x=\). Both coordinates. |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Graph] | B1 | Line \(y = 4-5x\) – must be straight, through approximately \((0,4)\) and intersecting the positive \(x\) axis near \((1,0)\) as shown. |
| [Graph] | B1 | Line \(y = \frac{4-x}{5}\) – must be straight and through approximately \((0, 0.8)\). No need to see intersection with \(x\) axis. |
| [Graph] | B1 | A line through \((0,0)\) and the point of intersection of a pair of straight lines with negative gradients. This line must be at \(45°\) unless scales are different in which case the line must be labelled \(y=x\). |
| 3 |
## Question 2(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{4-x}{5}$ | **B1** | OE |
| Equate a valid attempt at $f^{-1}$ with $f$, or with $x$, or $f$ with $x$ → $\left(\frac{2}{3}, \frac{2}{3}\right)$ or $(0.667, 0.667)$ | **M1, A1** | Equating and an attempt to solve as far as $x=$. Both coordinates. |
| | **3** | |
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## Question 2(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Graph] | **B1** | Line $y = 4-5x$ – must be straight, through approximately $(0,4)$ and intersecting the positive $x$ axis near $(1,0)$ as shown. |
| [Graph] | **B1** | Line $y = \frac{4-x}{5}$ – must be straight and through approximately $(0, 0.8)$. No need to see intersection with $x$ axis. |
| [Graph] | **B1** | A line through $(0,0)$ and the point of intersection of a pair of straight lines with negative gradients. This line must be at $45°$ unless scales are different in which case the line must be labelled $y=x$. |
| | **3** | |
---2 A function f is defined by $\mathrm { f } : x \mapsto 4 - 5 x$ for $x \in \mathbb { R }$.\\
(i) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and find the point of intersection of the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$.\\
(ii) Sketch, on the same diagram, the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$, making clear the relationship between the graphs.
\hfill \mbox{\textit{CAIE P1 2017 Q2 [6]}}