| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Standard +0.3 This is a straightforward composite function question requiring substitution to find constants, domain/range analysis, and finding an inverse. All steps follow standard procedures with no novel insight needed, making it slightly easier than average for A-level. |
| 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/Working | Marks | Guidance |
| \(2(ax^2+b)+3=6x^2-21\) | M1 | |
| \(a=3,\ b=-12\) | A1A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3x^2-12\geqslant 0\) or \(6x^2-21\geqslant 3\) | M1 | Allow \(=\) or \(\leqslant\) or \(>\) or \(<\). Ft from *their* \(a\), \(b\) |
| \(x\leqslant -2\) i.e. (max) \(q=-2\) | A1 [2] | Must be in terms of \(q\) (eg \(q\leqslant -2\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y\geqslant 6(-3)^2-21\Rightarrow\) range is \((y)\geqslant 33\) | B1 [1] | Do not allow \(y>33\). Accept all other notations e.g. \([33,\infty)\) or \([33,\infty]\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = 6x^2 - 21 \Rightarrow x = (\pm)\sqrt{\dfrac{y+21}{6}}\) | M1 | |
| \((fg)^{-1}(x) = -\sqrt{\dfrac{x+21}{6}}\) | A1 | Allow \(y = \ldots\) Must be a function of \(x\) |
| Domain is \(x \geqslant 33\) | B1√ | ft from *their* part (iii) but \(x\) essential |
| [3] |
## Question 10(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2(ax^2+b)+3=6x^2-21$ | M1 | |
| $a=3,\ b=-12$ | A1A1 [3] | |
## Question 10(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3x^2-12\geqslant 0$ or $6x^2-21\geqslant 3$ | M1 | Allow $=$ or $\leqslant$ or $>$ or $<$. Ft from *their* $a$, $b$ |
| $x\leqslant -2$ i.e. (max) $q=-2$ | A1 [2] | Must be in terms of $q$ (eg $q\leqslant -2$) |
## Question 10(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y\geqslant 6(-3)^2-21\Rightarrow$ range is $(y)\geqslant 33$ | B1 [1] | Do not allow $y>33$. Accept all other notations e.g. $[33,\infty)$ or $[33,\infty]$ |
## Question (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = 6x^2 - 21 \Rightarrow x = (\pm)\sqrt{\dfrac{y+21}{6}}$ | M1 | |
| $(fg)^{-1}(x) = -\sqrt{\dfrac{x+21}{6}}$ | A1 | Allow $y = \ldots$ Must be a function of $x$ |
| Domain is $x \geqslant 33$ | B1√ | ft from *their* part **(iii)** but $x$ essential |
| | [3] | |
---10 The function f is such that $\mathrm { f } ( x ) = 2 x + 3$ for $x \geqslant 0$. The function g is such that $\mathrm { g } ( x ) = a x ^ { 2 } + b$ for $x \leqslant q$, where $a , b$ and $q$ are constants. The function fg is such that $\operatorname { fg } ( x ) = 6 x ^ { 2 } - 21$ for $x \leqslant q$.\\
(i) Find the values of $a$ and $b$.\\
(ii) Find the greatest possible value of $q$.
It is now given that $q = - 3$.\\
(iii) Find the range of fg.\\
(iv) Find an expression for $( \mathrm { fg } ) ^ { - 1 } ( x )$ and state the domain of $( \mathrm { fg } ) ^ { - 1 }$.
\hfill \mbox{\textit{CAIE P1 2016 Q10 [9]}}