| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine domain for composite |
| Difficulty | Standard +0.3 This question tests understanding of composite function domains and basic algebraic manipulation. Part (i) requires recognizing that the range of f must lie within the domain of g, which is straightforward algebra. Parts (ii) and (iii) involve routine substitution and solving quadratic equations/inequalities with domain restrictions—standard techniques with no novel insight required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02u 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 |
| (i) \(3x + 1 \leqslant -1\) (Accept \(3x+1=-1\), \(3a+1=-1\)) | M1 | Do not allow gf in (i) to score in (iii) |
| \(x \leqslant -2/3 \Rightarrow\) largest value of \(a\) is \(-2/3\) (in terms of \(a\)) | A1 | Accept \(a \leqslant -2/3\) and \(a = -2/3\) |
| [2] | ||
| (ii) \(\text{fg}(x) = 3(1 - x^2) + 1\) | B1 | No marks in this part for gf used |
| \(\text{fg}(x) + 14 = 0 \Rightarrow 3x^2 = 12\) oe (2 terms) | B1 | |
| \(x = -2\) only | B1 | |
| [3] | ||
| (iii) \(\text{gf}(x) = -1 - (3x+1)^2\) oe | B1 | No marks in this part for fg used |
| \(\text{gf}(x) \leqslant -50 \Rightarrow (3x+1)^2 \geqslant 49\) (Allow \(\leqslant\) or \(=\)) | M1 | OR attempt soln of \(9x^2 + 6x - 48 + /\leqslant/\geqslant 0\) |
| \(3x+1 \geqslant 7\) or \(3x+1 \leqslant -7\) (one sufficient) www | A1 | |
| \(x \leqslant -8/3\) only www | A1 | OR \(x - 2 \geqslant\) *or* \(3x + 8 \leqslant 0\) (one suffic) |
| [4] |
## Question 8:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(i)** $3x + 1 \leqslant -1$ (Accept $3x+1=-1$, $3a+1=-1$) | M1 | Do not allow gf in (i) to score in (iii) |
| $x \leqslant -2/3 \Rightarrow$ largest value of $a$ is $-2/3$ (in terms of $a$) | A1 | Accept $a \leqslant -2/3$ and $a = -2/3$ |
| **[2]** | | |
| **(ii)** $\text{fg}(x) = 3(1 - x^2) + 1$ | B1 | No marks in this part for gf used |
| $\text{fg}(x) + 14 = 0 \Rightarrow 3x^2 = 12$ oe (2 terms) | B1 | |
| $x = -2$ only | B1 | |
| **[3]** | | |
| **(iii)** $\text{gf}(x) = -1 - (3x+1)^2$ oe | B1 | No marks in this part for fg used |
| $\text{gf}(x) \leqslant -50 \Rightarrow (3x+1)^2 \geqslant 49$ (Allow $\leqslant$ or $=$) | M1 | OR attempt soln of $9x^2 + 6x - 48 + /\leqslant/\geqslant 0$ |
| $3x+1 \geqslant 7$ or $3x+1 \leqslant -7$ (one sufficient) www | A1 | |
| $x \leqslant -8/3$ only www | A1 | OR $x - 2 \geqslant$ *or* $3x + 8 \leqslant 0$ (one suffic) |
| **[4]** | | |8 The function f is defined by $\mathrm { f } ( x ) = 3 x + 1$ for $x \leqslant a$, where $a$ is a constant. The function g is defined by $\mathrm { g } ( x ) = - 1 - x ^ { 2 }$ for $x \leqslant - 1$.\\
(i) Find the largest value of $a$ for which the composite function gf can be formed.
For the case where $a = - 1$,\\
(ii) solve the equation $\operatorname { fg } ( x ) + 14 = 0$,\\
(iii) find the set of values of $x$ which satisfy the inequality $\operatorname { gf } ( x ) \leqslant - 50$.
\hfill \mbox{\textit{CAIE P1 2015 Q8 [9]}}