| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation with inverses |
| Difficulty | Standard +0.3 This is a straightforward composite and inverse functions question requiring basic substitution for part (a) and solving a simple equation after finding two inverse functions in part (b). The inverses are linear and reciprocal forms that are standard, and the final equation solving is routine algebraic manipulation with no conceptual challenges. |
| 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 | Mark | Guidance |
| \(0\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((f^{-1}(x)) = \frac{x+2}{4}\), \((g^{-1}(x)) = \frac{4-x}{x}\) or \(\frac{4}{x} - 1\) | B1 B1 | OE. Sight of correct inverses. |
| \(x^2 + 6x - 16\ (= 0)\) | B1 | Equating inverses and simplifying. |
| \((x+8)\) and \((x-2)\) | M1 | Correct attempt at solution of *their* 3-term quadratic — factorising, completing the square or use of formula. |
| \((x =)\ 2\) or \(-8\) | A1 | Do not accept answers obtained with no method shown. |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0$ | B1 | |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(f^{-1}(x)) = \frac{x+2}{4}$, $(g^{-1}(x)) = \frac{4-x}{x}$ or $\frac{4}{x} - 1$ | B1 B1 | OE. Sight of correct inverses. |
| $x^2 + 6x - 16\ (= 0)$ | B1 | Equating inverses and simplifying. |
| $(x+8)$ and $(x-2)$ | M1 | Correct attempt at solution of *their* 3-term quadratic — factorising, completing the square or use of formula. |
| $(x =)\ 2$ or $-8$ | A1 | Do not accept answers obtained with no method shown. |
---5 Functions f and g are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) = 4 x - 2 , \quad \text { for } x \in \mathbb { R } , \\
& \mathrm {~g} ( x ) = \frac { 4 } { x + 1 } , \quad \text { for } x \in \mathbb { R } , x \neq - 1
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of fg (7).
\item Find the values of $x$ for which $\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q5 [6]}}