| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find intersection points |
| Difficulty | Standard +0.3 This is a standard C3 question on inverse functions with straightforward algebraic manipulation. Parts (a)-(c) are routine: stating range, finding inverse, and solving g(x)=x. Part (d) requires recognizing that g(a)=g^{-1}(a) implies a is the intersection point found in (c), which is a small conceptual step but well-signposted by 'hence'. Slightly easier than average due to the guided structure and standard techniques. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y \geqslant 3\) | B1 | Accept \(g(x) \geqslant 3\), Range \(\geqslant 3\), \([3,\infty)\); condone \(f \geqslant 3\); do not accept \(g(x) > 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 3 + \sqrt{x+2} \Rightarrow y - 3 = \sqrt{x+2} \Rightarrow x = (y-3)^2 - 2\) | M1 A1 | Attempt to make \(x\) subject; minimum expectation is 3 moved over and attempt to square; condone \(\sqrt{x+2} = y \pm 3 \Rightarrow x + 2 = y^2 \pm 9\) |
| \(g^{-1}(x) = (x-3)^2 - 2\), with \(x \geqslant 3\) | A1 | Correct function with correct domain or correct function with correct follow-through on range from (a); do not follow through on \(x \in \mathbb{R}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3 + \sqrt{x+2} = x \Rightarrow x + 2 = (x-3)^2 \Rightarrow x^2 - 7x + 7 = 0\) | M1, A1 | Sets equal, moves 3 over and squares both sides; \(= 0\) may be implied; can score for \(\sqrt{x+2} = x-3 \Rightarrow x+2 = x^2 \pm 9\) |
| \(x = \frac{7 \pm \sqrt{21}}{2}\) | M1, A1 | Correct method solving quadratic (must have real roots); allow decimal answers 5.79 and 1.21 |
| \(x = \frac{7 + \sqrt{21}}{2}\) only | Only positive root accepted; allow \(x = \frac{7}{2} + \sqrt{\frac{21}{4}}\) but not \(x = \frac{7 \pm \sqrt{21}}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = \frac{7 + \sqrt{21}}{2}\) | B1ft | Follow through on positive root from (c); may condone \(x = \frac{7+\sqrt{21}}{2}\); allow correct decimal awrt 5.79 |
## Question 3:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y \geqslant 3$ | B1 | Accept $g(x) \geqslant 3$, Range $\geqslant 3$, $[3,\infty)$; condone $f \geqslant 3$; do not accept $g(x) > 3$ |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 3 + \sqrt{x+2} \Rightarrow y - 3 = \sqrt{x+2} \Rightarrow x = (y-3)^2 - 2$ | M1 A1 | Attempt to make $x$ subject; minimum expectation is 3 moved over and attempt to square; condone $\sqrt{x+2} = y \pm 3 \Rightarrow x + 2 = y^2 \pm 9$ |
| $g^{-1}(x) = (x-3)^2 - 2$, with $x \geqslant 3$ | A1 | Correct function with correct domain **or** correct function with correct follow-through on range from (a); do not follow through on $x \in \mathbb{R}$ |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3 + \sqrt{x+2} = x \Rightarrow x + 2 = (x-3)^2 \Rightarrow x^2 - 7x + 7 = 0$ | M1, A1 | Sets equal, moves 3 over and squares both sides; $= 0$ may be implied; can score for $\sqrt{x+2} = x-3 \Rightarrow x+2 = x^2 \pm 9$ |
| $x = \frac{7 \pm \sqrt{21}}{2}$ | M1, A1 | Correct method solving quadratic (must have real roots); allow decimal answers 5.79 and 1.21 |
| $x = \frac{7 + \sqrt{21}}{2}$ only | | Only positive root accepted; allow $x = \frac{7}{2} + \sqrt{\frac{21}{4}}$ but **not** $x = \frac{7 \pm \sqrt{21}}{2}$ |
### Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = \frac{7 + \sqrt{21}}{2}$ | B1ft | Follow through on positive root from (c); may condone $x = \frac{7+\sqrt{21}}{2}$; allow correct decimal awrt 5.79 |
---3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-06_476_1107_221_422}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the graph of $y = \mathrm { g } ( x )$, where
$$g ( x ) = 3 + \sqrt { x + 2 } , \quad x \geqslant - 2$$
\begin{enumerate}[label=(\alph*)]
\item State the range of g .
\item Find $\mathrm { g } ^ { - 1 } ( x )$ and state its domain.
\item Find the exact value of $x$ for which
$$\mathrm { g } ( x ) = x$$
\item Hence state the value of $a$ for which
$$\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2017 Q3 [9]}}