| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation involving composites |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on composite and inverse functions requiring standard techniques: evaluating a composition (substitute and calculate), finding an inverse of a logarithmic function (swap and rearrange), and solving an equation involving a composition with an inverse (substitute the inverse, rearrange algebraically, and convert logarithm base). All parts are routine C3 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
3. The functions f and g are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } , \\
& \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } .
\end{aligned}$$
(i) Evaluate $\mathrm { gf } ( 1 )$.\\
(ii) Find an expression for $\mathrm { g } ^ { - 1 } ( x )$.\\
(iii) Find, in terms of natural logarithms, the solution of the equation
$$\mathrm { fg } ^ { - 1 } ( x ) = 2$$
\hfill \mbox{\textit{OCR C3 Q3 [8]}}