| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| 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 straightforward multi-part question testing standard techniques: completing the square (routine algebraic manipulation), finding range from completed square form, finding inverse of a quadratic with restricted domain, and function composition. All parts are textbook exercises requiring no problem-solving insight, though the multiple steps and careful handling of domain restrictions make it slightly more substantial than pure recall. |
| 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 |
|---|---|---|
| \(\{2(x-4)^2\}\ \{-9\}\) | B1 B1 | OE When \(a\) and \(b\) stated give priority to marking algebraic expression. |
| Answer | Marks | Guidance |
|---|---|---|
| \(y>-7\) | B1 | Allow \(f(x)>-7\) or \((-7,\infty)\). Don't allow \(x>-7\). |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-4)^2=\frac{y+9}{2}\) | M1 | 2 operations correct. Allow a sign error. |
| \(x=4\ [\pm]\sqrt{\frac{y+9}{2}}\) | M1 | 2 operations correct. Allow a sign error. |
| \([f^{-1}(x)=]4-\sqrt{\frac{x+9}{2}}\) | A1 FT | OE FT on *their* answer to (a) i.e. \(-a-\sqrt{\left(\frac{x-b}{2}\right)}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(fg(x)=f(2x+4)=2(2x+4-4)^2-9\) | M1 | Allow \(2(2x+4)^2-16(2x+4)+23\) |
| \(8x^2-9\) only | A1 |
## Question 6(a):
$\{2(x-4)^2\}\ \{-9\}$ | **B1 B1** | OE When $a$ and $b$ stated give priority to marking algebraic expression.
**2 total**
---
## Question 6(b):
$y>-7$ | **B1** | Allow $f(x)>-7$ or $(-7,\infty)$. Don't allow $x>-7$.
**1 total**
---
## Question 6(c):
$(x-4)^2=\frac{y+9}{2}$ | **M1** | 2 operations correct. Allow a sign error.
$x=4\ [\pm]\sqrt{\frac{y+9}{2}}$ | **M1** | 2 operations correct. Allow a sign error.
$[f^{-1}(x)=]4-\sqrt{\frac{x+9}{2}}$ | **A1 FT** | OE FT on *their* answer to **(a)** i.e. $-a-\sqrt{\left(\frac{x-b}{2}\right)}$
**3 total**
---
## Question 6(d):
$fg(x)=f(2x+4)=2(2x+4-4)^2-9$ | **M1** | Allow $2(2x+4)^2-16(2x+4)+23$
$8x^2-9$ only | **A1** |
**2 total**
---6 The function f is defined by $\mathrm { f } ( x ) = 2 x ^ { 2 } - 16 x + 23$ for $x < 3$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $2 ( x + a ) ^ { 2 } + b$.
\item Find the range of f.
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.\\
The function g is defined by $\mathrm { g } ( x ) = 2 x + 4$ for $x < - 1$.
\item Find and simplify an expression for $\mathrm { fg } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q6 [8]}}