| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2018 |
| Session | September |
| Marks | 8 |
| Topic | Composite & Inverse Functions |
| Type | Evaluate composite at point |
| Difficulty | Standard +0.3 This is a multi-part question on piecewise functions that requires finding constants from continuity conditions, evaluating a composite function, and explaining why an inverse doesn't exist. While it involves several steps, each part uses standard techniques (continuity at the boundary point, substitution for composition, and the horizontal line test). The composite evaluation ff(-1) is straightforward once the function is established. This is slightly easier than average as it's methodical rather than requiring insight. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| \(4a = 6 \Rightarrow a = \frac{3}{2}\) | M1 | Setting up equation with \(f(x)\) and \(-2\) |
| A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| \(\ln\left(\frac{3}{2}b - 8\right) - 2 = -2\) | M1* | Setting up equation with \(\ln(bx - 8) - 2 = -2\) and their value of \(a\) |
| \(\frac{3}{2}b - 8 = 1 \Rightarrow b = \ldots\) | dep*M1 | Correctly removing ln and solving for \(b\) |
| \(b = 6\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \ln 40 - 2\) | M1 | Indication of applying \(f(-1)\) to \(4 - 4x\) |
| A1 |
| Answer | Marks | Guidance |
|---|---|---|
| E.g. \(f\) is a many-one function – from the diagram there are, for example, two values of \(x\) for which \(f(x) = 0\) | B1 | oe – \(f\) is not one-one with evidence of why \(f\) is not one-one |
### (i)
$4 - 4a = -2$
$4a = 6 \Rightarrow a = \frac{3}{2}$ | M1 | Setting up equation with $f(x)$ and $-2$
| A1 | AG
**[2 marks total]**
### (ii)
$\ln\left(\frac{3}{2}b - 8\right) - 2 = -2$ | M1* | Setting up equation with $\ln(bx - 8) - 2 = -2$ and their value of $a$
$\frac{3}{2}b - 8 = 1 \Rightarrow b = \ldots$ | dep*M1 | Correctly removing ln and solving for $b$
$b = 6$ | A1 |
**[3 marks total]**
### (iii)
If $f(-1) = f(8)$
$= \ln 40 - 2$ | M1 | Indication of applying $f(-1)$ to $4 - 4x$
| A1 |
**[2 marks total]**
### (iv)
E.g. $f$ is a many-one function – from the diagram there are, for example, two values of $x$ for which $f(x) = 0$ | B1 | oe – $f$ is not one-one with evidence of why $f$ is not one-one
**[1 mark total]**
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-05_787_892_267_568}
The diagram shows the graph of $y = \mathrm { f } ( x )$, where
$$f ( x ) = \begin{cases} 4 - 4 x , & x \leqslant a , \\ \ln ( b x - 8 ) - 2 , & x \geqslant a . \end{cases}$$
The range of f is $\mathrm { f } ( x ) \geqslant - 2$.\\
(i) Show that $a = \frac { 3 } { 2 }$.\\
(ii) Find the value of $b$.\\
(iii) Find the exact value of $\mathrm { ff } ( - 1 )$.\\
(iv) Explain why the function f does not have an inverse.
\hfill \mbox{\textit{OCR H240/03 2018 Q4 [8]}}