| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Easy -1.3 This is a straightforward composite and inverse function question requiring only direct substitution and basic algebraic manipulation. All three parts involve routine procedures: (i) composing f with itself, (ii) applying three simple functions in sequence, and (iii) finding the inverse of a linear function. No problem-solving or conceptual insight is needed—just mechanical application of definitions. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(f^2(x) = 4x\) | B1 | |
| 1 | ||
| (ii) \(fgh(x) = fg(x+2)\) | M1 A1 | correct order of functions |
| \(= f(x+2)^2\) | A1 | |
| \(= 2(x+2)^2\) | ||
| 3 |
| Answer | Marks |
|---|---|
| \(\Rightarrow x = y-2\) | B1 |
| \(h^{-1}(x) = x-2\) | |
| 1 |
**(i)** $f^2(x) = 4x$ | B1 |
| 1 |
**(ii)** $fgh(x) = fg(x+2)$ | M1 A1 | correct order of functions
$= f(x+2)^2$ | A1 |
$= 2(x+2)^2$ | |
| 3 |
**(iii)** $y = h(x) = x+2$
$\Rightarrow x = y-2$ | B1 |
$h^{-1}(x) = x-2$ | |
| 1 |7 The functions $f , g$ and $h$ are defined as follows.
$$\mathrm { f } ( x ) = 2 x \quad \mathrm {~g} ( x ) = x ^ { 2 } \quad \mathrm {~h} ( x ) = x + 2$$
Find each of the following as functions of $x$.\\
(i) $\mathrm { f } ^ { 2 } ( x )$,\\
(ii) $\operatorname { fgh } ( x )$,\\
(iii) $\mathrm { h } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{OCR MEI C3 Q7 [5]}}