| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a straightforward composite and inverse function question with a simple square root function. Part (i) requires basic substitution (f(169)=16, then f(16)=7), part (ii) involves standard inverse function technique (swap x and y, rearrange to get y=(x-3)²), and part (iii) is routine graph sketching showing reflection in y=x. Slightly easier than average due to the simple function form and standard procedures required. |
| 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) Attempt correct process for composition | M1 | numerical or algebraic |
| Obtain \((16 \text{ and hence } 7)\) | A1 | 2 |
| (ii) Attempt correct process for finding inverse | M1 | maybe in terms of \(y\) so far |
| Obtain \((x - 3)^2\) | A1 | 2 or equiv; in terms of \(x\), not \(y\) |
| (iii) Sketch (more or less correct) \(y = f(x)\) | B1 | with 3 indicated or clearly implied on y-axis, correct curvature, no maximum point |
| Sketch (more or less correct) \(y = f^{-1}(x)\) | B1 | |
| State reflection in line \(y = x\) | B1 | 3 right hand half of parabola only or (explicit) equiv; independent of earlier marks |
(i) Attempt correct process for composition | M1 | numerical or algebraic
Obtain $(16 \text{ and hence } 7)$ | A1 | 2
(ii) Attempt correct process for finding inverse | M1 | maybe in terms of $y$ so far
Obtain $(x - 3)^2$ | A1 | 2 or equiv; in terms of $x$, not $y$
(iii) Sketch (more or less correct) $y = f(x)$ | B1 | with 3 indicated or clearly implied on y-axis, correct curvature, no maximum point
Sketch (more or less correct) $y = f^{-1}(x)$ | B1 |
State reflection in line $y = x$ | B1 | 3 right hand half of parabola only or (explicit) equiv; independent of earlier marks
---3 The function $f$ is defined for all non-negative values of $x$ by
$$f ( x ) = 3 + \sqrt { x }$$
(i) Evaluate ff(169).\\
(ii) Find an expression for $\mathrm { f } ^ { - 1 } ( \mathrm { x } )$ in terms of x .\\
(iii) On a single diagram sketch the graphs of $y = f ( x )$ and $y = f ^ { - 1 } ( x )$, indicating how the two graphs are related.
\hfill \mbox{\textit{OCR C3 2007 Q3 [7]}}