| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question on composite and inverse functions covering standard P1 content. Parts (a), (b), and (d) involve routine substitution and algebraic manipulation. Parts (c) and (e) test basic understanding of when inverses/composites exist using simple reasoning about domains and ranges. All parts are textbook-standard with no novel problem-solving required. |
| 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 | Mark | Guidance |
| \(a\left(x+\frac{1}{x}\right)+1\) | B1 | ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a\left(2+\frac{1}{2}\right)+1=11\) | M1 | Substitute \(x=2\) into their expression from (a) and equate to 11. May be done in 2 stages: \(f(2)=2.5,\ g(2.5)=11\) |
| \([a=]\ 4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| No, [because it is] not one-one | B1 | Or other suitable explanation including one to many or many to one |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([g^{-1}(x)]=\frac{x-1}{5}\) WWW | B1 | Condone use of \(a\) instead of 5 |
| \([g^{-1}f(x)=]\dfrac{x+\frac{1}{x}-1}{5}\) OE | M1 | Correct combination of their \(g^{-1}(x)\) with given \(f(x)\). Condone use of \(a\) instead of 5 |
| \(\frac{x^2-x+1}{5x}\) or \(\frac{1}{5}\left(x+\frac{1}{x}-1\right)\) or \(\frac{1}{5}(x+x^{-1}-1)\) OE ISW | A1 | Must not contain unresolved fractions e.g. \(\frac{x+x^{-1}-1}{5}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| The domain of f does not include the whole of the range of g. Or The range of g does not lie in the domain of f. | B1 | Accept answer that includes an example outside the domain of f, e.g. \(g(-1)=-4\) but for f, \(x>0\) |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $a\left(x+\frac{1}{x}\right)+1$ | B1 | ISW |
---
## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $a\left(2+\frac{1}{2}\right)+1=11$ | M1 | Substitute $x=2$ into their expression from (a) and equate to 11. May be done in 2 stages: $f(2)=2.5,\ g(2.5)=11$ |
| $[a=]\ 4$ | A1 | |
---
## Question 9(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| No, [because it is] not one-one | B1 | Or other suitable explanation including one to many or many to one |
---
## Question 9(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[g^{-1}(x)]=\frac{x-1}{5}$ WWW | B1 | Condone use of $a$ instead of 5 |
| $[g^{-1}f(x)=]\dfrac{x+\frac{1}{x}-1}{5}$ OE | M1 | Correct combination of their $g^{-1}(x)$ with given $f(x)$. Condone use of $a$ instead of 5 |
| $\frac{x^2-x+1}{5x}$ or $\frac{1}{5}\left(x+\frac{1}{x}-1\right)$ or $\frac{1}{5}(x+x^{-1}-1)$ OE ISW | A1 | Must not contain unresolved fractions e.g. $\frac{x+x^{-1}-1}{5}$ |
---
## Question 9(e):
| Answer | Mark | Guidance |
|--------|------|----------|
| The domain of f does not include the whole of the range of g. **Or** The range of g does not lie in the domain of f. | B1 | Accept answer that includes an example outside the domain of f, e.g. $g(-1)=-4$ but for f, $x>0$ |
---9 Functions f and g are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) = x + \frac { 1 } { x } \quad \text { for } x > 0 \\
& \mathrm {~g} ( x ) = a x + 1 \quad \text { for } x \in \mathbb { R }
\end{aligned}$$
where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\operatorname { gf } ( x )$.
\item Given that $\operatorname { gf } ( 2 ) = 11$, find the value of $a$.
\item Given that the graph of $y = \mathrm { f } ( x )$ has a minimum point when $x = 1$, explain whether or not f has an inverse.\\
It is given instead that $a = 5$.
\item Find and simplify an expression for $\mathrm { g } ^ { - 1 } \mathrm { f } ( x )$.
\item Explain why the composite function fg cannot be formed.\\
\includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-16_1143_1008_267_566}
The diagram shows a cross-section RASB of the body of an aircraft. The cross-section consists of a sector $O A R B$ of a circle of radius 2.5 m , with centre $O$, a sector $P A S B$ of another circle of radius 2.24 m with centre $P$ and a quadrilateral $O A P B$. Angle $A O B = \frac { 2 } { 3 } \pi$ and angle $A P B = \frac { 5 } { 6 } \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q9 [8]}}