| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a standard C3 inverse function question with multiple routine parts: completing the square (basic algebra), stating range (direct consequence), finding inverse (standard technique for quadratics with restricted domain), describing transformations (straightforward comparison), and finding a normal (requires derivative of inverse but uses standard methods). While multi-part with 5 sections, each step follows textbook procedures without requiring novel insight or particularly challenging problem-solving. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| \(f(x) = (x-1)^2 - 1 + 5 = (x-1)^2 + 4\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| \(f(x) \geq 4\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| \(y = (x-1)^2 + 4\) | ||
| \((x-1)^2 = y - 4\) | ||
| \(x - 1 = \pm\sqrt{y-4}\) | M1 | |
| \(x = 1 \pm \sqrt{y-4}\) | ||
| \(f^{-1}(x) = 1 + \sqrt{x-4}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| Translation by 4 units in negative \(x\) direction | B3 | |
| Translation by 1 unit in negative \(y\) direction (either first) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| \(\frac{dy}{dx} = \frac{1}{2}(x-4)^{-\frac{1}{2}}\) | M1 | |
| \(x = 8,\ y = 3,\ \text{grad} = \frac{1}{4}\) | A1 | |
| \(\therefore\) grad of normal \(= -4\) | ||
| \(\therefore y - 3 = -4(x-8)\) \([y = 35 - 4x]\) | M1 A1 | (12) |
# Question 8:
## Part (i)
| Answer | Mark | Notes |
|--------|------|-------|
| $f(x) = (x-1)^2 - 1 + 5 = (x-1)^2 + 4$ | M1 A1 | |
## Part (ii)
| Answer | Mark | Notes |
|--------|------|-------|
| $f(x) \geq 4$ | B1 | |
## Part (iii)
| Answer | Mark | Notes |
|--------|------|-------|
| $y = (x-1)^2 + 4$ | | |
| $(x-1)^2 = y - 4$ | | |
| $x - 1 = \pm\sqrt{y-4}$ | M1 | |
| $x = 1 \pm \sqrt{y-4}$ | | |
| $f^{-1}(x) = 1 + \sqrt{x-4}$ | A1 | |
## Part (iv)
| Answer | Mark | Notes |
|--------|------|-------|
| Translation by 4 units in negative $x$ direction | B3 | |
| Translation by 1 unit in negative $y$ direction (either first) | | |
## Part (v)
| Answer | Mark | Notes |
|--------|------|-------|
| $\frac{dy}{dx} = \frac{1}{2}(x-4)^{-\frac{1}{2}}$ | M1 | |
| $x = 8,\ y = 3,\ \text{grad} = \frac{1}{4}$ | A1 | |
| $\therefore$ grad of normal $= -4$ | | |
| $\therefore y - 3 = -4(x-8)$ $[y = 35 - 4x]$ | M1 A1 | **(12)** |
**Total: (72)**8. $f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1$.\\
(i) Express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$, where $a$ and $b$ are constants.\\
(ii) State the range of f .\\
(iii) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.\\
(iv) Describe fully two transformations that would map the graph of $y = \mathrm { f } ^ { - 1 } ( x )$ onto the graph of $y = \sqrt { x } , x \geq 0$.\\
(v) Find an equation for the normal to the curve $y = \mathrm { f } ^ { - 1 } ( x )$ at the point where $x = 8$.
\hfill \mbox{\textit{OCR C3 Q8 [12]}}