| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation with inverses |
| Difficulty | Standard +0.3 This is a standard C3 inverse functions question covering routine techniques: finding range, sketching reflections in y=x, deriving inverse algebraically, and solving f^(-1)(x)=g(x). Part (e) requires solving a quadratic equation but follows directly from substituting the expressions. All steps are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(x) \leq 3\) | B1 | |
| (b) [Graph showing \(y = f(x)\) and \(y = f^{-1}(x)\) reflected in a line, with appropriate shape] | B3 | |
| (c) \(y = 3 - x^2\) | M1 | |
| \(x^2 = 3 - y\) | M1 | |
| \(x = \pm\sqrt{3-y}\) | M1 | |
| \(f^{-1}(x) = \sqrt{3-x}, x \in \mathbb{R}, x \leq 3\) | M1 A2 | |
| (d) \(= f(\frac{4}{3}) = \frac{11}{9}\) | M1 A1 | |
| (e) \(\sqrt{3-x} = \frac{8}{3-x}\) | M1 | |
| \(3 - x = \frac{64}{(3-x)^2}\) | M1 | |
| \((3-x)^3 = 64\) | M1 | |
| \(3 - x = 4\) | M1 | |
| \(x = -1\) | A1 | (13) |
**(a)** $f(x) \leq 3$ | B1 |
**(b)** [Graph showing $y = f(x)$ and $y = f^{-1}(x)$ reflected in a line, with appropriate shape] | B3 |
**(c)** $y = 3 - x^2$ | M1 |
$x^2 = 3 - y$ | M1 |
$x = \pm\sqrt{3-y}$ | M1 |
$f^{-1}(x) = \sqrt{3-x}, x \in \mathbb{R}, x \leq 3$ | M1 A2 |
**(d)** $= f(\frac{4}{3}) = \frac{11}{9}$ | M1 A1 |
**(e)** $\sqrt{3-x} = \frac{8}{3-x}$ | M1 |
$3 - x = \frac{64}{(3-x)^2}$ | M1 |
$(3-x)^3 = 64$ | M1 |
$3 - x = 4$ | M1 |
$x = -1$ | A1 | (13)6. The function f is defined by
$$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$
\begin{enumerate}[label=(\alph*)]
\item State the range of f.
\item Sketch the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ on the same diagram.
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain.
The function g is defined by
$$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3 .$$
\item Evaluate $\mathrm { fg } ( - 3 )$.
\item Solve the equation
$$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x ) .$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q6 [13]}}