| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Easy -1.3 This is a straightforward question testing basic understanding of inverse functions and function composition. Part (i) requires only recalling that quadratic functions fail the horizontal line test (not one-to-one). Parts (ii) and (iii) are routine mechanical exercises: composing functions and finding the inverse of a linear function. All parts are standard textbook material with no problem-solving required. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
**Part (i)**
- It is a many-one function or equiv.: B1 **[1]**
**Part (ii)**
- Attempt to form $gf(x)$: M1
- Obtain $7x^2 - 2$ only: A1 **[2]**
**Part (iii)**
- Attempt to make $x$ the subject: M1
- Obtain $\frac{1}{7}(x + 2)$ only: A1 **[2]**
**[Total: 5]**5 Let $\mathrm { f } ( x ) = x ^ { 2 }$ and $\mathrm { g } ( x ) = 7 x - 2$ for all real values of $x$.\\
(i) Give a reason why f has no inverse function.\\
(ii) Write down an expression for $\mathrm { gf } ( x )$.\\
(iii) Find $\mathrm { g } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q5 [5]}}