| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Standard +0.3 This is a straightforward composite and inverse functions question requiring standard techniques: finding range from domain, inverting a rational function, composing functions, and checking if a composite is one-to-one. Part (iii) requires recognizing that f(x) > 5 makes the absolute value redundant, which is a small insight but routine for this topic. Overall slightly easier than average A-level as it's methodical application of standard procedures. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
11 The functions f and g are defined by $\mathrm { f } ( x ) = \frac { 1 } { 2 + x } + 5 , x > - 2$ and $\mathrm { g } ( x ) = | x | , x \in \mathbb { R }$.\\
(i) Given that the range of f is of the form $\mathrm { f } ( x ) > a$, find $a$.\\
(ii) Find an expression for $\mathrm { f } ^ { - 1 }$, stating its domain and range.\\
(iii) Show that $\mathrm { gf } ( x ) = \mathrm { f } ( x )$.\\
(iv) Find an expression for $\mathrm { fg } ( x )$. Determine whether fg has an inverse.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q11}}