| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on composite and inverse functions requiring standard techniques: finding an inverse function value (solve a cubic equation), composing two functions (substitute and simplify), and differentiating a composite function (chain rule). All parts are routine C3 exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Either: State \(2x^3 + 4 = -50\) | B1 | |
| State \(-3\) and no other | B1 | |
| Or: Obtain \(\sqrt[3]{\frac{1}{2}(x-4)}\) for inverse of f | B1 | or equiv; using any letter |
| State \(-3\) and no other | B1 | |
| Total | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show composition of functions the right way round | M1 | |
| Obtain \(2x - 16\) | A1 | AG; necessary detail needed. First step \(2(x-10)+4\) acceptable but then two more steps needed |
| Total | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain \(\sqrt[3]{2x^3 - 6}\) or \((2x^3-6)^{\frac{1}{3}}\) for \(\text{gf}(x)\) | B1 | or unsimplified equiv |
| Apply chain rule to function which is cube root of a non-linear expression | M1 | condone incorrect constant; otherwise use of chain rule for their function must be correct. May use \(u = 2x^3 - 6\); M1 earned for expression involving \(u\) … in terms of \(x\) |
| Obtain \(2x^2(2x^3-6)^{-\frac{2}{3}}\) | A1 | or similarly simplified equiv; do not accept final answer with \(\frac{6}{3}\) unsimplified |
| Total | [3] |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Either: State $2x^3 + 4 = -50$ | B1 | |
| State $-3$ and no other | B1 | |
| Or: Obtain $\sqrt[3]{\frac{1}{2}(x-4)}$ for inverse of f | B1 | or equiv; using any letter |
| State $-3$ and no other | B1 | |
| **Total** | **[2]** | |
---
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show composition of functions the right way round | M1 | |
| Obtain $2x - 16$ | A1 | AG; necessary detail needed. First step $2(x-10)+4$ acceptable but then two more steps needed |
| **Total** | **[2]** | |
---
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain $\sqrt[3]{2x^3 - 6}$ or $(2x^3-6)^{\frac{1}{3}}$ for $\text{gf}(x)$ | B1 | or unsimplified equiv |
| Apply chain rule to function which is cube root of a non-linear expression | M1 | condone incorrect constant; otherwise use of chain rule for their function must be correct. May use $u = 2x^3 - 6$; M1 earned for expression involving $u$ … in terms of $x$ |
| Obtain $2x^2(2x^3-6)^{-\frac{2}{3}}$ | A1 | or similarly simplified equiv; do not accept final answer with $\frac{6}{3}$ unsimplified |
| **Total** | **[3]** | |
---4 The functions f and g are defined for all real values of $x$ by
$$\mathrm { f } ( x ) = 2 x ^ { 3 } + 4 \quad \text { and } \quad \mathrm { g } ( x ) = \sqrt [ 3 ] { x - 10 }$$
(i) Evaluate $\mathrm { f } ^ { - 1 } ( - 50 )$.\\
(ii) Show that $\operatorname { fg } ( x ) = 2 x - 16$.\\
(iii) Differentiate $\operatorname { gf } ( x )$ with respect to $x$.
\hfill \mbox{\textit{OCR C3 2014 Q4 [7]}}