| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| 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 function question requiring standard techniques: substituting one function into another, simplifying algebraic fractions, finding an inverse by swapping x and y, and determining domain/range from the given constraints. 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 |
|---|---|---|
| \(fg(x) = 5x\) | M1A1 | |
| Range of \(fg\) is \(y \geqslant 0\) oe | B1 [3] | Accept \(y > 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 4/(5x+2) \Rightarrow x = (4-2y)/5y\) oe | M1 | Must be a function of \(x\) |
| \(g^{-1}(x) = (4-2x)/5x\) oe | A1 | |
| \(0, 2\) with no incorrect inequality | B1,B1 | |
| \(0 < x \leqslant 2\) oe, c.a.o. | B1 [5] |
## Question 8:
### Part (i):
$fg(x) = 5x$ | **M1A1** |
Range of $fg$ is $y \geqslant 0$ oe | **B1** [3] | Accept $y > 0$
### Part (ii):
$y = 4/(5x+2) \Rightarrow x = (4-2y)/5y$ oe | **M1** | Must be a function of $x$
$g^{-1}(x) = (4-2x)/5x$ oe | **A1** |
$0, 2$ with no incorrect inequality | **B1,B1** |
$0 < x \leqslant 2$ oe, c.a.o. | **B1** [5] |
---8 The functions $f$ and $g$ are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) = \frac { 4 } { x } - 2 \quad \text { for } x > 0 \\
& \mathrm {~g} ( x ) = \frac { 4 } { 5 x + 2 } \quad \text { for } x \geqslant 0
\end{aligned}$$
(i) Find and simplify an expression for $\mathrm { fg } ( x )$ and state the range of fg.\\
(ii) Find an expression for $\mathrm { g } ^ { - 1 } ( x )$ and find the domain of $\mathrm { g } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE P1 2016 Q8 [8]}}