| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Standard +0.3 This is a multi-part question covering standard C3 topics (quadratic functions, ranges, inverses, and composition). Part (i) requires completing the square (routine), part (ii) tests understanding of the horizontal line test, part (iii) involves algebraic manipulation of inverse functions (moderate), and part (iv) combines function composition with inequalities. While it requires multiple techniques, each step follows predictable patterns with no novel insight needed, making it slightly easier than the typical C3 question. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to find t-coord of staty point or complete square | M1 | — |
| Obtain \(\left(\frac{2}{3}, -9\right)\) or \(4\left(x - \frac{3}{2}\right)^2 - 9\) or \(-9\) | A1 | or equiv |
| State \(f(x) \ge -9\) | A1 3 | using any notation; with \(\ge\) |
| Answer | Marks | Guidance |
|---|---|---|
| Make one correct (perhaps general) relevant statement | B1 | not 1–1 , \(f\) is many-one, ...; maybe implied if attempt is specific to this f |
| Conclude with correct evidence related to this \(f\) | B1 2 AG | (more or less) correct sketch; correct relevant calculations, ... |
| Answer | Marks | Guidance |
|---|---|---|
| Either: Attempt to find expression for \(g^{-1}\) | *M1 | or equiv |
| Obtain \(\frac{1}{x}(x - b)\) | A1 | or equiv |
| Compare \(\frac{1}{a}(x - b)\) and \(ax + b\) | M1 | dep *M; by equating either coefficients of \(x\) or constant terms (or both); or substituting two non-zero values of \(x\) and solving eqns for \(a\) |
| Obtain at least \(-\frac{k}{a} = b\) and hence \(a = -1\) | A1 4 AG | necessary detail required; or equiv [SC1: first two steps as above, then substitute \(a = -1\): max possible M1A1B1] |
| [SC2: substitute \(a = -1\) at start: Attempt to find inverse | M1 | Obtain \(-x + b\) and conclude |
| Or: State or imply that \(y = g^{-1}(x)\) is reflection of \(y = g(x)\) in line \(y = x\) | B1 | — |
| State that line unchanged by this reflection is perpendicular to \(y = x\) | M2 | — |
| Conclude that \(a\) is \(-1\) | A1 4 | — |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply that \(gf(x) = -(4x^2 - 12x) + b\) | B1 | — |
| Attempt use of discriminant or relate to range of \(f\) | M1 | or equiv |
| Obtain \(64 + 16b < 0\) or \(9 + b < 5\) | A1 | or equiv |
| Obtain \(b < -4\) | A1 4 | — |
## (i)
Attempt to find t-coord of staty point or complete square | M1 | —
Obtain $\left(\frac{2}{3}, -9\right)$ or $4\left(x - \frac{3}{2}\right)^2 - 9$ or $-9$ | A1 | or equiv
State $f(x) \ge -9$ | A1 3 | using any notation; with $\ge$
## (ii)
Make one correct (perhaps general) relevant statement | B1 | not 1–1 , $f$ is many-one, ...; maybe implied if attempt is specific to this f
Conclude with correct evidence related to this $f$ | B1 2 AG | (more or less) correct sketch; correct relevant calculations, ...
## (iii)
Either: Attempt to find expression for $g^{-1}$ | *M1 | or equiv
Obtain $\frac{1}{x}(x - b)$ | A1 | or equiv
Compare $\frac{1}{a}(x - b)$ and $ax + b$ | M1 | dep *M; by equating either coefficients of $x$ or constant terms (or both); or substituting two non-zero values of $x$ and solving eqns for $a$
Obtain at least $-\frac{k}{a} = b$ and hence $a = -1$ | A1 4 AG | necessary detail required; or equiv [SC1: first two steps as above, then substitute $a = -1$: max possible M1A1B1]
[SC2: substitute $a = -1$ at start: Attempt to find inverse | M1 | Obtain $-x + b$ and conclude | A1 2]
Or: State or imply that $y = g^{-1}(x)$ is reflection of $y = g(x)$ in line $y = x$ | B1 | —
State that line unchanged by this reflection is perpendicular to $y = x$ | M2 | —
Conclude that $a$ is $-1$ | A1 4 | —
## (iv)
State or imply that $gf(x) = -(4x^2 - 12x) + b$ | B1 | —
Attempt use of discriminant or relate to range of $f$ | M1 | or equiv
Obtain $64 + 16b < 0$ or $9 + b < 5$ | A1 | or equiv
Obtain $b < -4$ | A1 4 | —
---The functions f and g are defined for all real values of $x$ by
$$f(x) = 4x^2 - 12x \quad \text{and} \quad g(x) = ax + b,$$
where $a$ and $b$ are non-zero constants.
\begin{enumerate}[label=(\roman*)]
\item Find the range of f. [3]
\item Explain why the function f has no inverse. [2]
\item Given that $g^{-1}(x) = g(x)$ for all values of $x$, show that $a = -1$. [4]
\item Given further that gf$(x) < 5$ for all values of $x$, find the set of possible values of $b$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2010 Q9 [13]}}