| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Topic | Composite & Inverse Functions |
| Type | State domain or range |
| Difficulty | Moderate -0.8 This is a straightforward composite and inverse functions question requiring routine techniques: stating ranges of simple functions, solving a quadratic equation from fg(x)=gf(x), and restricting a domain to make a function invertible. All parts are standard textbook exercises with no novel problem-solving required, making it easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
**(i)**
Range of f: $f(x) \geq 2$ **B1**
Range of g is all real numbers **B1**
**[2]**
**(ii)**
Obtain $(4x+3)^2 + 2$ and $4(x^2+2)+3$ **M1**
Obtain $16x^2 + 24x + 11 = 4x^2 + 11$ **A1**
Attempt to solve quadratic to obtain a value for $x$ **M1**
Obtain $x = 0$ and $x = -2$ **A1**
**[4]**
**(iii)**
Possibilities are $x \geq 0$ or $x \leq 0$ **B1**
Either $y = \sqrt{x-2}$ or $y = -\sqrt{x-2}$ as appropriate for the domain **B1\***
**[2]**7 The functions f and g are defined for all real numbers by
$$\mathrm { f } ( x ) = x ^ { 2 } + 2 \quad \text { and } \quad \mathrm { g } ( x ) = 4 x + 3$$
(i) State the range of each of the functions f and g .\\
(ii) Find the values of $x$ for which $\mathrm { fg } ( x ) = \mathrm { gf } ( x )$.\\
(iii) The function h , given by $\mathrm { h } ( x ) = x ^ { 2 } + 2$, has the same range as f but is such that $\mathrm { h } ^ { - 1 } ( x )$ exists. State a possible domain for h and find an expression for $\mathrm { h } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2016 Q7 [8]}}