| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on exponential functions covering standard C3 techniques: identifying range, finding inverse by swapping and rearranging, solving an exponential equation, and finding a tangent using differentiation. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations |
5. $\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , x \in \mathbb { R }$.\\
(i) State the range of f .\\
(ii) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain.\\
(iii) Solve the equation $\mathrm { f } ( x ) = 7$.\\
(iv) Find an equation for the tangent to the curve $y = \mathrm { f } ( x )$ at the point where $y = 7$.\\
\hfill \mbox{\textit{OCR C3 Q5 [10]}}