| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.3 This is a straightforward composite and inverse functions question requiring standard techniques: finding ranges by analyzing rational functions, composing functions algebraically, and finding an inverse by swapping and rearranging. All steps are routine for P1 level 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 |
|---|---|---|
| Answer | Marks | Guidance |
| Range of f is \(0 < f(x) < 3\) | B1B1 | OE. Range cannot be defined using \(x\) |
| Range of g is \(g(x) > 2\) | B1 | OE |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(fg(x) = \frac{3}{2(\frac{1}{x}+2)+1} = \frac{3x}{2+5x}\) | B1B1 | Second B mark implies first B mark |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = \frac{3x}{2+5x} \rightarrow 2y + 5xy = 3x \rightarrow 3x - 5xy = 2y\) | M1 | Correct order of operations |
| \(x(3-5y) = 2y \rightarrow x = \frac{2y}{3-5y}\) | M1 | Correct order of operations |
| \((fg)^{-1}(x) = \frac{2x}{3-5x}\) | A1 | |
| 3 |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Range of f is $0 < f(x) < 3$ | B1B1 | OE. Range cannot be defined using $x$ |
| Range of g is $g(x) > 2$ | B1 | OE |
| | **3** | |
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $fg(x) = \frac{3}{2(\frac{1}{x}+2)+1} = \frac{3x}{2+5x}$ | B1B1 | Second B mark implies first B mark |
| | **2** | |
## Question 7(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = \frac{3x}{2+5x} \rightarrow 2y + 5xy = 3x \rightarrow 3x - 5xy = 2y$ | M1 | Correct order of operations |
| $x(3-5y) = 2y \rightarrow x = \frac{2y}{3-5y}$ | M1 | Correct order of operations |
| $(fg)^{-1}(x) = \frac{2x}{3-5x}$ | A1 | |
| | **3** | |7 Functions f and g are defined by
$$\begin{aligned}
& \mathrm { f } : x \mapsto \frac { 3 } { 2 x + 1 } \quad \text { for } x > 0 \\
& \mathrm {~g} : x \mapsto \frac { 1 } { x } + 2 \quad \text { for } x > 0
\end{aligned}$$
(i) Find the range of f and the range of g .\\
(ii) Find an expression for $\mathrm { fg } ( x )$, giving your answer in the form $\frac { a x } { b x + c }$, where $a , b$ and $c$ are integers.\\
(iii) Find an expression for $( \mathrm { fg } ) ^ { - 1 } ( x )$, giving your answer in the same form as for part (ii).\\
\hfill \mbox{\textit{CAIE P1 2019 Q7 [8]}}