| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find minimum domain for inverse |
| Difficulty | Standard +0.3 This is a standard multi-part question on completing the square, finding domains for inverse functions, and composite functions. All parts follow routine procedures taught in P1 with no novel insights required. The completing the square and finding the turning point for one-one functions are textbook exercises, making this slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([2]\left[(x-3)^2\right][-7]\) | B1B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Largest value of \(k\) is \(3\). Allow \((k =) 3\). | B1 | Allow \(k \leqslant 3\) but not \(x \leqslant 3\) as final answer. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = 2(x-3)^2 - 7 \rightarrow (x-3)^2 = \frac{1}{2}(y+7)\) or with \(x/y\) transposed | M1 | Ft their \(a, b, c\). Order of operations correct. Allow sign errors |
| \(x = 3 \pm \sqrt{\frac{1}{2}(y+7)}\) Allow \(3 + \sqrt{\phantom{0}}\) or \(3 - \sqrt{\phantom{0}}\) or with \(x/y\) transposed | DM1 | Ft their \(a, b, c\). Order of operations correct. Allow sign errors |
| \(f^{-1}(x) = 3 - \sqrt{\frac{1}{2}(x+7)}\) | A1 | |
| (Domain is \(x\)) \(\geqslant\) their \(-7\) | B1FT | Allow other forms for interval but if variable appears must be \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x + 3 \leqslant 1\). Allow \(x + 3 = 1\) | M1 | Allow \(x + 3 \leqslant k\) |
| Largest \(p\) is \(-2\). Allow \((p =) -2\) | A1 | Allow \(p \leqslant -2\) but not \(x \leqslant -2\) as final answer. |
| \(\text{fg}(x) = \text{f}(x+3) = 2x^2 - 7\) cao | B1 |
## Question 11(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[2]\left[(x-3)^2\right][-7]$ | B1B1B1 | |
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## Question 11(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Largest value of $k$ is $3$. Allow $(k =) 3$. | B1 | Allow $k \leqslant 3$ but not $x \leqslant 3$ as final answer. |
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## Question 11(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = 2(x-3)^2 - 7 \rightarrow (x-3)^2 = \frac{1}{2}(y+7)$ or with $x/y$ transposed | M1 | Ft their $a, b, c$. Order of operations correct. Allow sign errors |
| $x = 3 \pm \sqrt{\frac{1}{2}(y+7)}$ Allow $3 + \sqrt{\phantom{0}}$ or $3 - \sqrt{\phantom{0}}$ or with $x/y$ transposed | DM1 | Ft their $a, b, c$. Order of operations correct. Allow sign errors |
| $f^{-1}(x) = 3 - \sqrt{\frac{1}{2}(x+7)}$ | A1 | |
| (Domain is $x$) $\geqslant$ their $-7$ | B1FT | Allow other forms for interval but if variable appears must be $x$ |
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## Question 11(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x + 3 \leqslant 1$. Allow $x + 3 = 1$ | M1 | Allow $x + 3 \leqslant k$ |
| Largest $p$ is $-2$. Allow $(p =) -2$ | A1 | Allow $p \leqslant -2$ but not $x \leqslant -2$ as final answer. |
| $\text{fg}(x) = \text{f}(x+3) = 2x^2 - 7$ cao | B1 | |11 (i) Express $2 x ^ { 2 } - 12 x + 11$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants.\\
The function f is defined by $\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 11$ for $x \leqslant k$.\\
(ii) State the largest value of the constant $k$ for which f is a one-one function.\\
(iii) For this value of $k$ find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$.\\
The function g is defined by $\mathrm { g } ( x ) = x + 3$ for $x \leqslant p$.\\
(iv) With $k$ now taking the value 1 , find the largest value of the constant $p$ which allows the composite function fg to be formed, and find an expression for $\mathrm { fg } ( x )$ whenever this composite function exists.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P1 2018 Q11 [11]}}