| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2015 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Sketch function and inverse graphs |
| Difficulty | Challenging +1.2 This is a structured multi-part question on a standard A-level topic (inverse functions) with straightforward calculus and algebraic manipulation. Part (a) requires routine differentiation and solving, parts (b-c) test understanding of domain/range relationships, part (d) involves standard inverse function algebra (solving a quadratic), and part (e) combines the results. While it requires careful work across multiple parts and some algebraic fluency, each individual step uses familiar techniques without requiring novel insight or particularly complex reasoning. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02m Graphs of functions: difference between plotting and sketching1.02v Inverse and composite functions: graphs and conditions for existence1.07n Stationary points: find maxima, minima using derivatives |
% Figure shows a curve with maximum at point A, passing through origin O, with horizontal asymptote
\includegraphics{figure_1}
Figure 1 shows a sketch of the curve with equation $y = f(x)$ where
$$f(x) = \frac{x^2 + 16}{3x} \quad x \neq 0$$
The curve has a maximum at the point $A$ with coordinates $(a, b)$.
(a) Find the value of $a$ and the value of $b$.
[4]
The function g is defined as
$$g : x \mapsto \frac{x^2 + 16}{3x} \quad a \leq x < 0$$
where $a$ is the value found in part (a).
(b) Write down the range of g.
[1]
(c) On the same axes sketch $y = g(x)$ and $y = g^{-1}(x)$.
[3]
(d) Find an expression for $g^{-1}(x)$ and state the domain of $g^{-1}$
[5]
(e) Solve the equation $g(x) = g^{-1}(x)$.
[3]
\hfill \mbox{\textit{Edexcel AEA 2015 Q5 [16]}}