| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.3 This is a standard multi-part question on composite and inverse functions requiring completing the square, finding inverse functions, and composite functions. All techniques are routine A-level procedures with no novel problem-solving required, though the multiple parts and domain/range considerations make it slightly more substantial than the most basic exercises. |
| 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 |
|---|---|
| 9(a): \([(x-2)^2]\ [-1]\) | B1 B1 |
| 9(b): Smallest \(c = 2\) (FT on their part (a)) | B1FT |
| 9(c): \(y = (x-2)^2 - 1 \rightarrow (x-2)^2 = y+1\) | *M1 |
| \(x = 2(\pm)\sqrt{y+1}\) | DM1 |
| \((f^{-1}(x)) = 2 + \sqrt{x+1}\) for \(x > 8\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{gf}(x) = \frac{1}{(x-2)^2 - 1 + 1} = \frac{1}{(x-2)^2}\) | B1 | OE |
| Range of gf is \(0 < \text{gf}(x) < \frac{1}{9}\) | B1 B1 |
## Question 9:
**9(a):** $[(x-2)^2]\ [-1]$ | B1 B1 |
**9(b):** Smallest $c = 2$ (FT on their part (a)) | B1FT |
**9(c):** $y = (x-2)^2 - 1 \rightarrow (x-2)^2 = y+1$ | *M1 |
$x = 2(\pm)\sqrt{y+1}$ | DM1 |
$(f^{-1}(x)) = 2 + \sqrt{x+1}$ for $x > 8$ | A1 |
## Question 9(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{gf}(x) = \frac{1}{(x-2)^2 - 1 + 1} = \frac{1}{(x-2)^2}$ | B1 | OE |
| Range of gf is $0 < \text{gf}(x) < \frac{1}{9}$ | B1 B1 | |
---9 The functions f and g are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 3 \text { for } x > c , \text { where } c \text { is a constant, } \\
& \mathrm { g } ( x ) = \frac { 1 } { x + 1 } \quad \text { for } x > - 1
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $( x - a ) ^ { 2 } + b$.\\
It is given that f is a one-one function.
\item State the smallest possible value of $c$.\\
It is now given that $c = 5$.
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$.
\item Find an expression for $\mathrm { gf } ( x )$ and state the range of gf .
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q9 [9]}}