| 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 +0.2 This is a standard C3 functions question covering routine techniques: finding range from a restricted quadratic, sketching a function and its inverse as reflections in y=x, finding inverse by swapping and rearranging, and evaluating composite functions. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than the average A-level question. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.02z Models in context: use functions in modelling |
| Answer | Marks |
|---|---|
| \(f(x) \le 3\) | B1 |
| Answer | Marks |
|---|---|
| Graph showing \(y = f(x)\) and \(y = f^{-1}(x)\) with correct reflection about \(y = x\) | B3 |
| Answer | Marks |
|---|---|
| \(x^2 = 3 - y\), \(x = \pm\sqrt{3-y}\) | M1 |
| \(f^{-1}(x) = \sqrt{3-x}\), \(x \in \mathbb{R}\), \(x \le 3\) | A2 |
| Answer | Marks |
|---|---|
| \(=f\left(\frac{4}{3}\right) = \frac{11}{9}\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = -1\) | M1, M1, M1, A1 | (12) |
## (i)
$f(x) \le 3$ | B1 |
## (ii)
Graph showing $y = f(x)$ and $y = f^{-1}(x)$ with correct reflection about $y = x$ | B3 |
## (iii)
$y = 3 - x^2$
$x^2 = 3 - y$, $x = \pm\sqrt{3-y}$ | M1 |
$f^{-1}(x) = \sqrt{3-x}$, $x \in \mathbb{R}$, $x \le 3$ | A2 |
## (iv)
$=f\left(\frac{4}{3}\right) = \frac{11}{9}$ | M1 A1 |
## (v)
$\sqrt{3-x} = \frac{8}{3-x}$
$3 - x = \frac{64}{(3-x)^2}$
$(3-x)^3 = 64$
$3 - x = 4$
$x = -1$ | M1, M1, M1, A1 | (12)
---The function f is defined by
$$\text{f}(x) \equiv 3 - x^2, \quad x \in \mathbb{R}, \quad x \geq 0.$$
\begin{enumerate}[label=(\roman*)]
\item State the range of f. [1]
\item Sketch the graphs of $y = \text{f}(x)$ and $y = \text{f}^{-1}(x)$ on the same diagram. [3]
\item Find an expression for f$^{-1}(x)$ and state its domain. [3]
\end{enumerate}
The function g is defined by
$$\text{g}(x) \equiv \frac{8}{3-x}, \quad x \in \mathbb{R}, \quad x \neq 3.$$
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Evaluate fg$(-3)$. [2]
\item Solve the equation
$$\text{f}^{-1}(x) = \text{g}(x).$$ [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q8 [12]}}