| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Topic | Composite & Inverse Functions |
| Type | State domain or range |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on function basics. Finding the range of a simple root function, evaluating a composition, finding an inverse by swapping and rearranging, and sketching reflection in y=x are all standard techniques requiring minimal problem-solving. Slightly easier than average due to the simple function form, but the multiple parts bring it close to typical difficulty. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
**Question 2:**
**(i)** $\text{f}(x) \geqslant 4$ **B1** (Accept $y \geqslant 4$ but not $x$)
**(ii)** $16$ **B1**
**(iii)** $\left(\frac{y-4}{3}\right)^2 = x$ **M1**
$\text{f}^{-1}(x) = \frac{(x-4)^2}{9}$ **A1** (Accept $y = \text{f}(x)$)
**(iv)** [Graph showing f and its inverse] **B2**
- **B1** for general shape of $y = \text{f}(x)$ starting at $(0, 4)$
- **B1** for general shape of inverse starting at approximately $(4, 0)$ and a suggestion of intersection
State: Reflection in line $y = x$ **B1**
**Total: 7 marks**2 It is given that $\mathrm { f } ( x ) = 4 + 3 \sqrt { x }$, where $x \geqslant 0$.\\
(i) State the range of f .\\
(ii) State the value of $\mathrm { ff } ( 16 )$.\\
(iii) Find $\mathrm { f } ^ { - 1 } ( x )$.\\
(iv) On the same axes, sketch the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ and state how the graphs are related.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2018 Q2 [7]}}