| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find function from composite |
| Difficulty | Standard +0.3 This is a straightforward composite function question requiring standard techniques: finding k from a range condition (completing the square or using vertex form), determining range of fg (substitution and analysis), and finding h(x) from gh(x) using inverse functions. All steps are routine A-level procedures with no novel insight 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 existence1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to form expression for \(\text{gf}(x)\) | \*M1 | Expect \(5\big((3x-2)^2+k\big)-1\); \(\text{fg}(x)\) is M0. Do not allow algebraic errors. |
| Obtain \(5(3x-2)^2+5k-1\) | A1 | OE e.g. \(45x^2-60x+5k+19\) |
| *Their* \(5k-1=39\) or \(5k-1 \geqslant 39\) | DM1 | Or use \(b^2-4ac=0\) (must be \('=0'\), could be implied later) on \(45x^2-60x+5k+19-39 \geqslant 0\) OE. |
| Obtain \(k=8\) | A1 | Do not accept \(k \geqslant 8\) |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtaining \(\big(3(5x-1)-2\big)^2 + \text{their } k\) | M1 | May simplify and/or use \(k\) at this stage; \(k\) may have come from an inequality in (a). |
| Conclude \([\text{fg}(x)] \geqslant 8\) allow \([y] \geqslant 8\) | A1 FT | OE. Following *their* value of \(k\); must be \(\geqslant\), not \(>\). Allow an accurate written description. |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(g^{-1}(x) = \frac{1}{5}(x+1)\) | B1 | OE. \(\frac{1}{5}(x+1)\) must be indicated as the inverse. |
| \([h(x)=]7x+4\) | B1 B1 | If \(7x+4\) only, it must be clear that this is \(h(x)\). |
| Total: 3 |
## Question 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to form expression for $\text{gf}(x)$ | \*M1 | Expect $5\big((3x-2)^2+k\big)-1$; $\text{fg}(x)$ is M0. Do not allow algebraic errors. |
| Obtain $5(3x-2)^2+5k-1$ | A1 | OE e.g. $45x^2-60x+5k+19$ |
| *Their* $5k-1=39$ or $5k-1 \geqslant 39$ | DM1 | Or use $b^2-4ac=0$ (must be $'=0'$, could be implied later) on $45x^2-60x+5k+19-39 \geqslant 0$ OE. |
| Obtain $k=8$ | A1 | Do not accept $k \geqslant 8$ |
| **Total: 4** | | |
---
## Question 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtaining $\big(3(5x-1)-2\big)^2 + \text{their } k$ | M1 | May simplify and/or use $k$ at this stage; $k$ may have come from an inequality in **(a)**. |
| Conclude $[\text{fg}(x)] \geqslant 8$ allow $[y] \geqslant 8$ | A1 FT | OE. Following *their* value of $k$; must be $\geqslant$, not $>$. Allow an accurate written description. |
| **Total: 2** | | |
---
## Question 9(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $g^{-1}(x) = \frac{1}{5}(x+1)$ | B1 | OE. $\frac{1}{5}(x+1)$ must be indicated as the inverse. |
| $[h(x)=]7x+4$ | B1 B1 | If $7x+4$ only, it must be clear that this is $h(x)$. |
| **Total: 3** | | |9 The functions f and g are defined for all real values of $x$ by
$$f ( x ) = ( 3 x - 2 ) ^ { 2 } + k \quad \text { and } \quad g ( x ) = 5 x - 1$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Given that the range of the function gf is $\mathrm { gf } ( x ) \geqslant 39$, find the value of $k$.
\item For this value of $k$, determine the range of the function fg .
\item The function h is defined for all real values of $x$ and is such that $\mathrm { gh } ( x ) = 35 x + 19$. Find an expression for $\mathrm { g } ^ { - 1 } ( x )$ and hence, or otherwise, find an expression for $\mathrm { h } ( x )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-12_739_625_260_721}
The diagram shows the circle with centre $C ( - 4,5 )$ and radius $\sqrt { 20 }$ units. The circle intersects the $y$-axis at the points $A$ and $B$. The size of angle $A C B$ is $\theta$ radians.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q9 [9]}}