| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.3 This is a straightforward composite and inverse function question requiring standard techniques: substituting one function into another, then finding the inverse by swapping x and y. The domain/range analysis is routine for rational functions. Slightly easier than average as it involves no novel problem-solving, just methodical application of well-practiced procedures. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence |
**(i)**
Compose correctly gf to give $\dfrac{3}{x-1} + 2\left(= \dfrac{2x+1}{x-1}\right)$ [B1]
State domain $x \neq 1$ or equivalent notation [dep B1]
State range $y \neq 2$ or equivalent notation [dep B1]
**Subtotal: [3]**
**(ii)**
Attempt correct method to find inverse [M1]
State $(y=) \dfrac{x+1}{x-2}$ or $(y=) \dfrac{3}{x-2} + 1$ [A1]
State $x = 2$ [depA1]
**Subtotal: [3]**
**Total: [6]**6 The functions f and g are given by $\mathrm { f } ( x ) = \frac { 3 } { x - 1 }$ for all $x \neq 1$ and $\mathrm { g } ( x ) = x + 2$ for all real $x$.\\
(i) Find gf, stating its domain and range.\\
(ii) Find $( \mathrm { gf } ) ^ { - 1 }$, stating any values of $x$ for which $( \mathrm { gf } ) ^ { - 1 }$ is not defined.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2015 Q6 [6]}}