| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Verify composite identity |
| Difficulty | Moderate -0.3 This is a straightforward composite function question requiring routine application of standard techniques: identifying ranges (trivial for exponential functions), substituting one function into another, and differentiating using chain rule. Part (b) is a 'show that' verification rather than independent derivation, and part (d) involves basic differentiation and solving a simple equation. All parts are textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| \(g(x) \geq 1\) | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(f \circ g(x) = f(e^{x^2}) = 3e^{x^2} + \ln e^{x^2}\) | M1 | |
| \(= x^2 + 3e^{x^2}\) | cso | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(fg(x) \geq 3\) | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d}{dx}(x^2 + 3e^{x^2}) = 2x + 6xe^{x^2}\) | M1 A1 | |
| \(2x + 6xe^{x^2} = x^2e^{x^2} + 2x\) | M1 | |
| \(e^{x^2}(6x - x^2) = 0\) | M1 | |
| \(e^{x^2} \neq 0\), \(6x - x^2 = 0\) | A1 | |
| \(x = 0, 6\) | A1 A1 | (6) [10] |
**(a)**
$g(x) \geq 1$ | B1 | (1)
**(b)**
$f \circ g(x) = f(e^{x^2}) = 3e^{x^2} + \ln e^{x^2}$ | M1
$= x^2 + 3e^{x^2}$ | cso | A1 | (2)
(fg: $x \mapsto x^2 + 3e^{x^2}$)
**(c)**
$fg(x) \geq 3$ | B1 | (1)
**(d)**
$\frac{d}{dx}(x^2 + 3e^{x^2}) = 2x + 6xe^{x^2}$ | M1 A1
$2x + 6xe^{x^2} = x^2e^{x^2} + 2x$ | M1
$e^{x^2}(6x - x^2) = 0$ | M1
$e^{x^2} \neq 0$, $6x - x^2 = 0$ | A1
$x = 0, 6$ | A1 A1 | (6) [10]
---5. The functions $f$ and $g$ are defined by
$$\begin{aligned}
& \mathrm { f } : x \mapsto 3 x + \ln x , \quad x > 0 , \quad x \in \mathbb { R } \\
& \mathrm {~g} : x \mapsto \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R }
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Write down the range of g.
\item Show that the composite function fg is defined by
$$\mathrm { fg } : x \mapsto x ^ { 2 } + 3 \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } .$$
\item Write down the range of fg.
\item Solve the equation $\frac { \mathrm { d } } { \mathrm { d } x } [ \mathrm { fg } ( x ) ] = x \left( x \mathrm { e } ^ { x ^ { 2 } } + 2 \right)$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2009 Q5 [10]}}