| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find function from composite |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on standard P1 topics: completing the square (routine), finding an inverse function (standard procedure), and finding a function from a composite (algebraic manipulation). All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((x-2)^2\) | M1 | Must be "\(-2\)" \(\pm k\) |
| \((x-2)^2 + 3\) | A1 | co |
| \(f(x) > 3\) | B1\(\sqrt{}\) | ft on their '3' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x - 2 = (\pm)\sqrt{y-3}\) | M1 | \(\pm\) not required for M mark |
| \(f^{-1}(x) = 2 + \sqrt{x-3}\) | A1 | \(f(x)\) + removal of minus sign needed |
| domain is \(x > 3\) | B1\(\sqrt{}\) | ft domain of \(f^{-1}\) = range of \(f\) or for \(f^{-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(h(x) = x^2 + 3\) | B1 | co |
## Question 7:
**Part (i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(x-2)^2$ | M1 | Must be "$-2$" $\pm k$ |
| $(x-2)^2 + 3$ | A1 | co |
| $f(x) > 3$ | B1$\sqrt{}$ | ft on their '3' |
**Part (ii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x - 2 = (\pm)\sqrt{y-3}$ | M1 | $\pm$ not required for M mark |
| $f^{-1}(x) = 2 + \sqrt{x-3}$ | A1 | $f(x)$ + removal of minus sign needed |
| domain is $x > 3$ | B1$\sqrt{}$ | ft domain of $f^{-1}$ = range of $f$ or for $f^{-1}$ |
**Part (iii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $h(x) = x^2 + 3$ | B1 | co |
---7 The function f is defined by
$$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7 \text { for } x > 2$$
(i) Express $\mathrm { f } ( x )$ in the form $( x - a ) ^ { 2 } + b$ and hence state the range of f .\\
(ii) Obtain an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$.
The function g is defined by
$$\mathrm { g } ( x ) = x - 2 \text { for } x > 2$$
The function h is such that $\mathrm { f } = \mathrm { hg }$ and the domain of h is $x > 0$.\\
(iii) Obtain an expression for $\mathrm { h } ( x )$.
\hfill \mbox{\textit{CAIE P1 2010 Q7 [7]}}