| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2021 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Evaluate composite at point |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard function operations: identifying range of a quadratic (simple transformation), evaluating a composite function at a point (substitute and calculate), and finding an inverse of a rational function (standard algebraic manipulation). All parts are routine textbook exercises requiring no problem-solving insight, though slightly more involved than pure recall due to the rational function algebra. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| \(y \leqslant 7\) | B1 | Allow f\((x)\) or f for \(y\). Allow e.g. \(\{y \in \mathbb{R}: y \leqslant 7\}\), \(-\infty < y \leqslant 7\), \((-\infty, 7]\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(1.8) = 7 - 2 \times 1.8^2 = 0.52 \Rightarrow\) gf\((1.8) = g(0.52) = \frac{3 \times 0.52}{5 \times 0.52 - 1} = \ldots\) | M1 | Full method to find f(1.8) and substitute into g. Also allow substituting \(x = 1.8\) into gf\((x)\) |
| gf\((1.8) = 0.975\) or e.g. \(\frac{39}{40}\) | A1 | Correct value |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = \frac{3x}{5x-1} \Rightarrow 5xy - y = 3x \Rightarrow x(5y-3) = y\) | M1 | Correct attempt to cross multiply, then factorise \(x\) from an \(xy\) term and an \(x\) term |
| \(g^{-1}(x) = \frac{x}{5x-3}\) | A1 | Allow equivalent correct expressions e.g. \(\frac{-x}{3-5x}\), \(\frac{1}{5} + \frac{3}{25x-15}\). Ignore any domain if given. |
## Question 2:
### Part (a):
$y \leqslant 7$ | B1 | Allow f$(x)$ or f for $y$. Allow e.g. $\{y \in \mathbb{R}: y \leqslant 7\}$, $-\infty < y \leqslant 7$, $(-\infty, 7]$
### Part (b):
$f(1.8) = 7 - 2 \times 1.8^2 = 0.52 \Rightarrow$ gf$(1.8) = g(0.52) = \frac{3 \times 0.52}{5 \times 0.52 - 1} = \ldots$ | M1 | Full method to find f(1.8) and substitute into g. Also allow substituting $x = 1.8$ into gf$(x)$
gf$(1.8) = 0.975$ or e.g. $\frac{39}{40}$ | A1 | Correct value
### Part (c):
$y = \frac{3x}{5x-1} \Rightarrow 5xy - y = 3x \Rightarrow x(5y-3) = y$ | M1 | Correct attempt to cross multiply, then factorise $x$ from an $xy$ term and an $x$ term
$g^{-1}(x) = \frac{x}{5x-3}$ | A1 | Allow equivalent correct expressions e.g. $\frac{-x}{3-5x}$, $\frac{1}{5} + \frac{3}{25x-15}$. Ignore any domain if given.
---\begin{enumerate}
\item The functions f and g are defined by
\end{enumerate}
$$\begin{aligned}
& f ( x ) = 7 - 2 x ^ { 2 } \quad x \in \mathbb { R } \\
& \operatorname { g } ( x ) = \frac { 3 x } { 5 x - 1 } \quad x \in \mathbb { R } \quad x \neq \frac { 1 } { 5 }
\end{aligned}$$
(a) State the range of f\\
(b) Find gf (1.8)\\
(c) Find $\mathrm { g } ^ { - 1 } ( x )$
\hfill \mbox{\textit{Edexcel Paper 2 2021 Q2 [5]}}