| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.3 This is a standard multi-part question on completing the square and inverse functions. Part (i) is routine algebraic manipulation, parts (ii)-(iii) test understanding of range and one-one functions from vertex form, and part (iv) requires finding an inverse function. All techniques are textbook exercises with no novel problem-solving required, making it 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/Working | Marks | Guidance |
| \((x-2)^2 - 4 + k\) | B1B1 [2] | \(a = -2\), \(b = -4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) > k - 4\) or \([k-4, \infty)\) or \((k-4, \infty)\) oe | B1 [follow through] [1] | ft *their* \(k - 4\). Accept \(>\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Smallest value of \(p = 2\) | B1 [follow through] [1] | ft *their* 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x - 2 = (\pm)\sqrt{y + 4 - k}\) | M1 | |
| \(x = 2 + \sqrt{y + 4 - k}\) | A1 [follow through] | ft from *their* part (i) |
| \(f^{-1}(x) = 2 + \sqrt{x + 4 - k}\) | A1 | cao |
| Domain is \(x > k - 4\) or \([k-4, \infty)\) or \((k-4, \infty)\) oe | B1 [follow through] [4] | ft from *their* part (ii). Accept \(>\) |
## Question 8:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(x-2)^2 - 4 + k$ | B1B1 [2] | $a = -2$, $b = -4$ |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) > k - 4$ or $[k-4, \infty)$ or $(k-4, \infty)$ oe | B1 [follow through] [1] | ft *their* $k - 4$. Accept $>$ |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Smallest value of $p = 2$ | B1 [follow through] [1] | ft *their* 2 |
### Part (iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x - 2 = (\pm)\sqrt{y + 4 - k}$ | M1 | |
| $x = 2 + \sqrt{y + 4 - k}$ | A1 [follow through] | ft from *their* part (i) |
| $f^{-1}(x) = 2 + \sqrt{x + 4 - k}$ | A1 | cao |
| Domain is $x > k - 4$ or $[k-4, \infty)$ or $(k-4, \infty)$ oe | B1 [follow through] [4] | ft from *their* part (ii). Accept $>$ |
---8 The function $\mathrm { f } : x \mapsto x ^ { 2 } - 4 x + k$ is defined for the domain $x \geqslant p$, where $k$ and $p$ are constants.\\
(i) Express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b + k$, where $a$ and $b$ are constants.\\
(ii) State the range of f in terms of $k$.\\
(iii) State the smallest value of $p$ for which f is one-one.\\
(iv) For the value of $p$ found in part (iii), find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain $\mathrm { f } ^ { - 1 }$, giving your answers in terms of $k$.
\hfill \mbox{\textit{CAIE P1 2012 Q8 [8]}}