| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2017 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Standard +0.3 This is a standard C3/C4 composite and inverse functions question with routine algebraic manipulation. Parts (a)-(c) are textbook exercises requiring direct application of standard techniques. Part (d) adds mild problem-solving by requiring students to form and solve an equation, but the algebra is straightforward and the 'show that' format provides the answer. Slightly above average due to the multi-part nature and part (d), but still a familiar question type. |
| Spec | 1.02u 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 |
| \(0 < f(x) < \frac{4}{5}\) | M1 A1 (2) | M1: One correct limit. A1: Fully correct; accept \(0 < y < \frac{4}{5}\), \((0, 0.8)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = \frac{4}{3x+5} \Rightarrow (3x+5)y = 4\) | M1 | Set \(y=f(x)\) or \(x=f(y)\) and multiply by denominator |
| \(\Rightarrow x = \frac{4-5y}{3y}\) | dM1 | Make \(x\) (or swapped \(y\)) subject; condone arithmetic slips |
| \(f^{-1}(x) = \frac{4-5x}{3x}\) with domain \(\left(0 < x < \frac{4}{5}\right)\) | A1 o.e. (3) | Domain not required for A mark; ISW after correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(fg(x) = \frac{4}{\frac{3}{x}+5}\) | B1 (1) | Allow any correct form then ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{3x+5}{4} = \frac{4}{\frac{3}{x}+5}\) | M1 | Sets \(fg(x)=gf(x)\) with both sides correct |
| \(15x^2 + 18x + 15 = 0\) | A1 | Correct 3TQ |
| Uses \(18^2 < 4\times15\times15\) and deduces no real roots | M1 A1 (4) | M1: Attempt discriminant. A1: Correct work, \(b^2-4ac = -576\), and conclusion |
# Question 4:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0 < f(x) < \frac{4}{5}$ | M1 A1 (2) | M1: One correct limit. A1: Fully correct; accept $0 < y < \frac{4}{5}$, $(0, 0.8)$ |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = \frac{4}{3x+5} \Rightarrow (3x+5)y = 4$ | M1 | Set $y=f(x)$ or $x=f(y)$ and multiply by denominator |
| $\Rightarrow x = \frac{4-5y}{3y}$ | dM1 | Make $x$ (or swapped $y$) subject; condone arithmetic slips |
| $f^{-1}(x) = \frac{4-5x}{3x}$ with domain $\left(0 < x < \frac{4}{5}\right)$ | A1 o.e. (3) | Domain not required for A mark; ISW after correct answer |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $fg(x) = \frac{4}{\frac{3}{x}+5}$ | B1 (1) | Allow any correct form then ISW |
## Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{3x+5}{4} = \frac{4}{\frac{3}{x}+5}$ | M1 | Sets $fg(x)=gf(x)$ with both sides correct |
| $15x^2 + 18x + 15 = 0$ | A1 | Correct 3TQ |
| Uses $18^2 < 4\times15\times15$ and deduces no real roots | M1 A1 (4) | M1: Attempt discriminant. A1: Correct work, $b^2-4ac = -576$, and conclusion |
---\begin{enumerate}
\item Given that
\end{enumerate}
$$\begin{array} { l l }
\mathrm { f } ( x ) = \frac { 4 } { 3 x + 5 } , & x > 0 \\
\mathrm {~g} ( x ) = \frac { 1 } { x } , & x > 0
\end{array}$$
(a) state the range of f,\\
(b) find $\mathrm { f } ^ { - 1 } ( x )$,\\
(c) find $\mathrm { fg } ( x )$.\\
(d) Show that the equation $\mathrm { fg } ( x ) = \mathrm { gf } ( x )$ has no real solutions.
\hfill \mbox{\textit{Edexcel C34 2017 Q4 [10]}}