| Exam Board | Edexcel |
|---|---|
| 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), stating range (direct consequence), finding inverse (standard technique), describing transformations (straightforward), and finding a normal (requires derivative of inverse but follows standard method). While multi-part with 5 sections, each component uses well-practiced techniques without requiring novel insight, making it slightly easier than average overall. |
| 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)1.07m Tangents and normals: gradient and equations |
| Answer | Marks |
|---|---|
| (a) \(f(x) = (x-1)^2-1+5 = (x-1)^2+4\) | M1 A1 |
| (b) \(f(x) \geq 4\) | B1 |
| Answer | Marks |
|---|---|
| \(x-1 = \pm\sqrt{y-4}\) | M1 |
| Answer | Marks |
|---|---|
| \(f^{-1}(x) = 1+\sqrt{x-4}\) | M1 A1 |
| Answer | Marks |
|---|---|
| Translation by 1 unit in negative \(y\) direction (either first) | B2 |
| (e) \(\frac{dy}{dx} = \frac{1}{2}(x-4)^{-\frac{1}{2}}\) | M1 |
| \(x = 8, y = 3\), grad \(= \frac{1}{2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\therefore y-3 = -4(x-8)\) | [\(y = 35-4x\)] | M1 A1 |
**(a)** $f(x) = (x-1)^2-1+5 = (x-1)^2+4$ | M1 A1
**(b)** $f(x) \geq 4$ | B1
**(c)** $y = (x-1)^2+4$
$(x-1)^2 = y-4$
$x-1 = \pm\sqrt{y-4}$ | M1
$x = 1 \pm \sqrt{y-4}$
$f^{-1}(x) = 1+\sqrt{x-4}$ | M1 A1
**(d)** Translation by 4 units in negative $x$ direction
Translation by 1 unit in negative $y$ direction (either first) | B2
**(e)** $\frac{dy}{dx} = \frac{1}{2}(x-4)^{-\frac{1}{2}}$ | M1
$x = 8, y = 3$, grad $= \frac{1}{2}$ | A1
$\therefore$ grad of normal $= -4$
$\therefore y-3 = -4(x-8)$ | [$y = 35-4x$] | M1 A1 | **(12)**7. $\quad f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$, where $a$ and $b$ are constants.
\item State the range of f.
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
\item Describe fully two transformations that would map the graph of $y = \mathrm { f } ^ { - 1 } ( x )$ onto the graph of $y = \sqrt { x } , x \geq 0$.
\item Find an equation for the normal to the curve $y = \mathrm { f } ^ { - 1 } ( x )$ at the point where $x = 8$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q7 [12]}}