| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine domain for composite |
| Difficulty | Standard +0.3 This is a straightforward composite/inverse functions question requiring basic understanding of domains and ranges. Part (i) is routine, part (ii) involves simple substitution and recognizing domain restrictions, and part (iii) requires finding when g(x) falls outside f's range—all standard C3 techniques with no novel problem-solving required. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.05a Sine, cosine, tangent: definitions for all arguments |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(-2 \leq y \leq 2\) | B1 | allow \(<\); any notation |
| State \(y \leq 4\) | B1 | 2 allow \(<\); any notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show correct process for composition | M1 | right way round |
| Obtain or imply 0.959 and hence 2.16 | A1 | AG; necessary detail required |
| Obtain \(g(0.5) = 3.5\) | B1 | or (unsimplified) equiv |
| Observe that 3.5 not in domain of f | B1 | 4 or equiv |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Relate quadratic expression to at least one end of range of f | M1 | or equiv |
| Obtain both of \(4-2x^2 < -2\) and \(4-2x^2 > 2\) | A1 | or equiv; allow any sign in each (\(<\) or \(\leq\) or \(>\) or \(\geq\) or \(=\)) |
| Obtain at least two of the \(x\) values \(-\sqrt{3}, -1, 1, \sqrt{3}\) | A1 | |
| Obtain all four of the \(x\) values | A1 | |
| Attempt solution involving four \(x\) values | M1 | to produce at least two sets of values |
| Obtain \(x < -\sqrt{3},\ -1 < x < 1,\ x > \sqrt{3}\) | A1 | 6 allow \(\leq\) instead of \(<\) and/or \(\geq\) instead of \(>\) |
# Question 9:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $-2 \leq y \leq 2$ | B1 | allow $<$; any notation |
| State $y \leq 4$ | B1 | **2** allow $<$; any notation |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show correct process for composition | M1 | right way round |
| Obtain or imply 0.959 and hence 2.16 | A1 | AG; necessary detail required |
| Obtain $g(0.5) = 3.5$ | B1 | or (unsimplified) equiv |
| Observe that 3.5 not in domain of f | B1 | **4** or equiv |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Relate quadratic expression to at least one end of range of f | M1 | or equiv |
| Obtain both of $4-2x^2 < -2$ and $4-2x^2 > 2$ | A1 | or equiv; allow any sign in each ($<$ or $\leq$ or $>$ or $\geq$ or $=$) |
| Obtain at least two of the $x$ values $-\sqrt{3}, -1, 1, \sqrt{3}$ | A1 | |
| Obtain all four of the $x$ values | A1 | |
| Attempt solution involving four $x$ values | M1 | to produce at least two sets of values |
| Obtain $x < -\sqrt{3},\ -1 < x < 1,\ x > \sqrt{3}$ | A1 | **6** allow $\leq$ instead of $<$ and/or $\geq$ instead of $>$ |9 Functions $f$ and $g$ are defined by
$$\begin{array} { l l }
\mathrm { f } ( x ) = 2 \sin x & \text { for } - \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi \\
\mathrm {~g} ( x ) = 4 - 2 x ^ { 2 } & \text { for } x \in \mathbb { R } .
\end{array}$$
(i) State the range of f and the range of g .\\
(ii) Show that $\operatorname { gf } ( 0.5 ) = 2.16$, correct to 3 significant figures, and explain why $\mathrm { fg } ( 0.5 )$ is not defined.\\
(iii) Find the set of values of $x$ for which $\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )$ is not defined.
\hfill \mbox{\textit{OCR C3 2007 Q9 [12]}}