| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a straightforward C3 inverse function question requiring standard techniques: evaluating a composite function (routine substitution), finding an inverse by swapping x and y then solving (standard method for a quadratic with restricted domain), and sketching by reflecting in y=x. All steps are textbook procedures 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 existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either: Obtain \(f(-3) = -7\) | B1 | maybe implied |
| Show correct process for compn of functions | M1 | |
| Obtain \(-47\) | A1 3 | |
| Or: Show correct process for compn of functions | M1 | using algebraic approach |
| Obtain \(2 - (2 - x^2)^2\) | A1 | or equiv |
| Obtain \(-47\) | A1 (3) | |
| (ii) Attempt correct process for finding inverse | M1 | as far as \(x = \ldots\) or equiv |
| Obtain either one of \(x = \pm\sqrt{2-y}\) or both | A1 | or equiv perhaps involving \(x\) |
| Obtain correct \(-\sqrt{2-x}\) | A1 3 | or equiv; in terms of \(x\) now |
| (iii) Draw graph showing attempt at reflection in \(y = x\) | M1 | |
| Draw (more or less) correct graph | A1 | with end-point on x-axis and no minimum point in third quadrant |
| Indicate coordinates 2 and \(-\sqrt{2}\) | A1 3 | accept \(-1.4\) in place of \(-\sqrt{2}\) |
**(i)** Either: Obtain $f(-3) = -7$ | B1 | maybe implied
Show correct process for compn of functions | M1 |
Obtain $-47$ | A1 3 |
Or: Show correct process for compn of functions | M1 | using algebraic approach
Obtain $2 - (2 - x^2)^2$ | A1 | or equiv
Obtain $-47$ | A1 (3) |
**(ii)** Attempt correct process for finding inverse | M1 | as far as $x = \ldots$ or equiv
Obtain either one of $x = \pm\sqrt{2-y}$ or both | A1 | or equiv perhaps involving $x$
Obtain correct $-\sqrt{2-x}$ | A1 3 | or equiv; in terms of $x$ now
**(iii)** Draw graph showing attempt at reflection in $y = x$ | M1 |
Draw (more or less) correct graph | A1 | with end-point on x-axis and no minimum point in third quadrant
Indicate coordinates 2 and $-\sqrt{2}$ | A1 3 | accept $-1.4$ in place of $-\sqrt{2}$6\\
\includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-3_563_583_267_781}
The diagram shows the graph of $y = \mathrm { f } ( x )$, where
$$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$
(i) Evaluate $\mathrm { ff } ( - 3 )$.\\
(ii) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.\\
(iii) Sketch the graph of $y = \mathrm { f } ^ { - 1 } ( x )$. Indicate the coordinates of the points where the graph meets the axes.
\hfill \mbox{\textit{OCR C3 2006 Q6 [9]}}