| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Topic | Function Transformations |
| Type | Composite transformation sketch |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard transformations and inverse functions. Part (i) requires sketching a cubic and recognizing the horizontal line test fails. Part (ii) is trivial algebra. Part (iii) involves applying transformations to sketch and counting intersections visually. All techniques are routine with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Sketch should show a maximum touching at \(x = 0\) and crossing the axis at \(x = 2\) | B1 | |
| \(f\) is many-one over part of its domain OR the graph of \(f\) doubles back on itself OR is not one-one OR a horizontal line crosses the graph more than once. | B1 | |
| \(g^{-1}(x) = \frac{1}{2}x + \frac{1}{2}\) | [2] | |
| (ii) Clear intention to show their \(f(x)\) moved to the right and down | B1 | [1] |
| (iii) Show a straight line with a positive gradient. States "one root" | Dep B1, Dep B1 | [3] |
(i) Sketch should show a maximum touching at $x = 0$ and crossing the axis at $x = 2$ | B1 |
$f$ is many-one over part of its domain OR the graph of $f$ doubles back on itself OR is not one-one OR a horizontal line crosses the graph more than once. | B1 |
$g^{-1}(x) = \frac{1}{2}x + \frac{1}{2}$ | | [2]
(ii) Clear intention to show their $f(x)$ moved to the right and down | B1 | [1]
(iii) Show a straight line with a positive gradient. States "one root" | Dep B1, Dep B1 | [3]Let $f(x) = x^2(x - 2)$ and $g(x) = 2x - 1$ for all real $x$.
\begin{enumerate}[label=(\roman*)]
\item Sketch the graph of $y = f(x)$ and explain briefly why the function f has no inverse. [2]
\item Write down $g^{-1}(x)$. [1]
\item On the same diagram, sketch the graphs of $y = f(x - 1) - 3$ and $y = g^{-1}(x)$ and state the number of real roots of the equation $f(x - 1) - 3 = g^{-1}(x)$. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2010 Q3 [6]}}