| Exam Board | AQA |
|---|---|
| Module | Paper 1 (Paper 1) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a standard A-level functions question covering range, domain, inverse functions, and composition. All parts follow routine procedures: finding range of an exponential function, stating domain of inverse, algebraic manipulation to find inverse, composing functions, and solving an exponential equation. While it requires multiple techniques across 10 marks, each individual step is textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| 10(a) | Makes a deduction about the | |
| lower bound of the function (4) | AO2.2a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| using set notation | AO2.5 | B1 |
| (b)(i) | States correctly the set they | |
| gave in part (a) | AO1.2 | B1F |
| (b)(ii) | Interchanges x and y at any | |
| stage | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Rearranges and takes logs | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| throughout | AO1.1b | A1 |
| (c)(i) | Obtains gf(x) | AO1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| (c)(ii) | Forms equation and rearranges | |
| using ‘their’ gf(x)= 2 | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| requires the use of logs.) | AO1.1b | A1F |
| Obtains correct solution | AO1.1b | A1 |
| Total | 10 | |
| Q | Marking Instructions | AO |
Question 10:
--- 10(a) ---
10(a) | Makes a deduction about the
lower bound of the function (4) | AO2.2a | B1 | The range of f is the set
x:x4,x
Correctly states the range of f
using set notation | AO2.5 | B1
(b)(i) | States correctly the set they
gave in part (a) | AO1.2 | B1F | x:x4,x
(b)(ii) | Interchanges x and y at any
stage | AO1.1a | M1 | y = 4 + 3 –x
x = 4 + 3 –y
3 –y = x – 4
–y = log (x – 4)
3
f –1(x) = –log (x – 4)
3
Rearranges and takes logs | AO1.1a | M1
Obtains correct expression from
completely correct working for
f1(x), notation correct
throughout | AO1.1b | A1
(c)(i) | Obtains gf(x) | AO1.1b | B1 | gf(x) = g(4 + 3 –x )
= 5 – (4 + 3 –x )0.5
(c)(ii) | Forms equation and rearranges
using ‘their’ gf(x)= 2 | AO1.1a | M1 | 5 – (4 + 3 –x )0.5 = 2
(4 + 3–x ) = 9
3 –x = 5
x = – log 5
3
Correctly rearranges to get a
single exponential term where
logs can be taken. (Follow
through provided ‘their’ equation
requires the use of logs.) | AO1.1b | A1F
Obtains correct solution | AO1.1b | A1
Total | 10
Q | Marking Instructions | AO | Marks | Typical SolutionThe function f is defined by
$$f(x) = 4 + 3^{-x}, \quad x \in \mathbb{R}$$
\begin{enumerate}[label=(\alph*)]
\item Using set notation, state the range of f
[2 marks]
\item The inverse of f is $f^{-1}$
\begin{enumerate}[label=(\roman*)]
\item Using set notation, state the domain of $f^{-1}$
[1 mark]
\item Find an expression for $f^{-1}(x)$
[3 marks]
\end{enumerate}
\item The function g is defined by
$$g(x) = 5 - \sqrt{x}, \quad (x \in \mathbb{R} : x > 0)$$
\begin{enumerate}[label=(\roman*)]
\item Find an expression for $gf(x)$
[1 mark]
\item Solve the equation $gf(x) = 2$, giving your answer in an exact form.
[3 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 1 Q10 [10]}}