| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.3 This is a straightforward composite and inverse functions question requiring standard techniques: finding range of f to determine domain of g (simple substitution of endpoints), composing two simple functions, and inverting a rational function. All steps are routine for P1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| 4(i) | Max(a) is 8 | B1 |
| Min(b) is 24 | B1 | Allow b = 24 or b(cid:46)24 |
| 2 | SCB1 for 8 and 24 seen |
| Answer | Marks |
|---|---|
| 4(ii) | 96 100−4x |
| Answer | Marks | Guidance |
|---|---|---|
| x−1 x−1 | B1 | 48 |
| Answer | Marks |
|---|---|
| 4(iii) | 96 96 96 |
| Answer | Marks | Guidance |
|---|---|---|
| x−1 x−1 y+4 | M1 | FT from their(ii) provided (ii) involves algebraic fraction. |
| Answer | Marks | Guidance |
|---|---|---|
| x+4 | A1 | 100+x |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | Max(a) is 8 | B1 | Allow a = 8 or a(cid:45)8
Min(b) is 24 | B1 | Allow b = 24 or b(cid:46)24
2 | SCB1 for 8 and 24 seen
--- 4(ii) ---
4(ii) | 96 100−4x
gf(x) = −4 or gf(x) =
x−1 x−1 | B1 | 48
2 −4 is insufficient
x−1
Apply ISW
1
--- 4(iii) ---
4(iii) | 96 96 96
y= −4 → y+4= → x−1=
x−1 x−1 y+4 | M1 | FT from their(ii) provided (ii) involves algebraic fraction.
Allow sign errors
( gf )−1( x )= 96 +1
x+4 | A1 | 100+x
OR . Must be a function of x. Apply ISW
x+4
2
Question | Answer | Marks | GuidanceThe function f is defined by $\text{f}(x) = \frac{48}{x - 1}$ for $3 \leqslant x \leqslant 7$. The function g is defined by $\text{g}(x) = 2x - 4$ for $a \leqslant x \leqslant b$, where $a$ and $b$ are constants.
\begin{enumerate}[label=(\roman*)]
\item Find the greatest value of $a$ and the least value of $b$ which will permit the formation of the composite function gf. [2]
It is now given that the conditions for the formation of gf are satisfied.
\item Find an expression for $\text{gf}(x)$. [1]
\item Find an expression for $(\text{gf})^{-1}(x)$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2019 Q4 [5]}}