| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.8 This is a straightforward composite and inverse functions question requiring standard techniques: substitution for fg(x), algebraic manipulation to find the inverse, and solving an equation involving two composites. All steps are routine for P1 level with no novel problem-solving required, making it easier than average. |
| 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 | Marks | Guidance |
| \(fg(x) = (2x+1)^2 + 3\) | B1 | OE |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = (2x+1)^2 + 3 \rightarrow 2x+1 = (\pm)\sqrt{y-3}\) | M1 | 1st two operations. Allow one sign error or \(x/y\) interchanged |
| \(x = (\pm)\frac{1}{2}\left(\sqrt{y-3} - 1\right)\) | M1 | OE 2nd two operations. Allow one sign error or \(x/y\) interchanged |
| \(\left(fg^{-1}(x) =\right) \frac{1}{2}\left(\sqrt{x-3} - 1\right)\) for \((x) > 3\) | A1 B1 | Allow \((3, \infty)\) |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(gf(x) = 2(x^2+3)+1\) | B1 | SOI |
| \((2x+1)^2 + 3 - 3 = 2(x^2+3)+1 \rightarrow 2x^2 + 4x - 6\ (=0)\) | *M1 | Express as 3-term quadratic |
| \((2)(x+3)(x-1)\ (=0)\) | DM1 | Or quadratic formula or completing the square |
| \(x = 1\) | A1 | |
| Total | 4 |
## Question 11(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $fg(x) = (2x+1)^2 + 3$ | B1 | OE |
| **Total** | **1** | |
## Question 11(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = (2x+1)^2 + 3 \rightarrow 2x+1 = (\pm)\sqrt{y-3}$ | M1 | 1st two operations. Allow one sign error or $x/y$ interchanged |
| $x = (\pm)\frac{1}{2}\left(\sqrt{y-3} - 1\right)$ | M1 | OE 2nd two operations. Allow one sign error or $x/y$ interchanged |
| $\left(fg^{-1}(x) =\right) \frac{1}{2}\left(\sqrt{x-3} - 1\right)$ for $(x) > 3$ | A1 B1 | Allow $(3, \infty)$ |
| **Total** | **4** | |
## Question 11(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $gf(x) = 2(x^2+3)+1$ | B1 | SOI |
| $(2x+1)^2 + 3 - 3 = 2(x^2+3)+1 \rightarrow 2x^2 + 4x - 6\ (=0)$ | *M1 | Express as 3-term quadratic |
| $(2)(x+3)(x-1)\ (=0)$ | DM1 | Or quadratic formula or completing the square |
| $x = 1$ | A1 | |
| **Total** | **4** | |11 The functions $f$ and $g$ are defined by
$$\begin{array} { l l }
f ( x ) = x ^ { 2 } + 3 & \text { for } x > 0 \\
g ( x ) = 2 x + 1 & \text { for } x > - \frac { 1 } { 2 }
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\mathrm { fg } ( x )$.
\item Find an expression for $( \mathrm { fg } ) ^ { - 1 } ( x )$ and state the domain of $( \mathrm { fg } ) ^ { - 1 }$.
\item Solve the equation $\mathrm { fg } ( x ) - 3 = \mathrm { gf } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q11 [9]}}