CAIE P1 2024 November — Question 5 10 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComposite & Inverse Functions
TypeFind inverse function
DifficultyModerate -0.3 This is a standard P1 composite and inverse functions question with routine techniques: evaluating a function, sketching an inverse (reflection in y=x), finding an inverse algebraically, and solving an equation involving function composition. All parts follow textbook methods with no novel insight required, making it slightly easier than average.
Spec1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence
The function f is defined by \(\mathrm{f}(x) = \frac{2x + 1}{2x - 1}\) for \(x < \frac{1}{2}\).
    1. State the value of f\((-1)\). [1]
    2. \includegraphics{figure_5} The diagram shows the graph of \(y = \mathrm{f}(x)\). Sketch the graph of \(y = \mathrm{f}^{-1}(x)\) on this diagram. Show any relevant mirror line. [2]
    3. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain of the function \(\mathrm{f}^{-1}\). [4]
The function g is defined by \(\mathrm{g}(x) = 3x + 2\) for \(x \in \mathbb{R}\).
  1. Solve the equation \(\mathrm{f}(x) = \mathrm{gf}\left(\frac{1}{4}\right)\). [3]
Question 5:
AnswerMarks
5(a)(iii)2x+1
= y  2x+1= y(2x−1)
AnswerMarks Guidance
2x−1M1* Equating y to the given function and clearing of fractions.
x and y may be interchanged at this stage.
AnswerMarks Guidance
2xy−2x= y+1DM1 Condone  errors during simplification.
x+1 −x−1
,
AnswerMarks Guidance
2(x−1) 2−2xA1 Allow ‘f−1’or ‘y =’ but NOT ‘x =’, nor fractions within
fractions.
AnswerMarks Guidance
[Domain of f−1 is] x1B1 Accept — ∞ < x <1 or (— ∞, 1), condone [— ∞, 1).
Alternative Method for Question 5(a)(iii)
2 2
y=1+  y−1=
AnswerMarks Guidance
2x−1 2x−1M1* Equating y to the given function after division by 2x−1.
Isolating the term in x.
x and y may be interchanged at this stage.
2
2x= +1
AnswerMarks Guidance
y−1DM1 Condone  errors during simplification.
1 1
+
AnswerMarks Guidance
x−1 2A1 OE
Allow ‘f−1’or ‘y =’ but NOT ‘x =’, nor fractions within
fractions.
AnswerMarks Guidance
[Domain of f−1 is] x1B1 Accept — ∞ < x <1 or (— ∞, 1), condone [— ∞, 1).
4
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
5(b)1
gf  =−7
AnswerMarks
4B1
2x+1
= −7
AnswerMarks Guidance
2x−1M1 2x+1 1
Equating to their gf  .
2x−1 4
3
x=
AnswerMarks Guidance
8A1 OE
Alternative solution for Question 5(b)
1
gf  =−7
AnswerMarks Guidance
4B1
x= f −1(−7)M1  1
x= f −1 their gf  
 4
3
x=
AnswerMarks Guidance
8A1 OE
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 5:
--- 5(a)(iii) ---
5(a)(iii) | 2x+1
= y  2x+1= y(2x−1)
2x−1 | M1* | Equating y to the given function and clearing of fractions.
x and y may be interchanged at this stage.
2xy−2x= y+1 | DM1 | Condone  errors during simplification.
x+1 −x−1
,
2(x−1) 2−2x | A1 | Allow ‘f−1’or ‘y =’ but NOT ‘x =’, nor fractions within
fractions.
[Domain of f−1 is] x1 | B1 | Accept — ∞ < x <1 or (— ∞, 1), condone [— ∞, 1).
Alternative Method for Question 5(a)(iii)
2 2
y=1+  y−1=
2x−1 2x−1 | M1* | Equating y to the given function after division by 2x−1.
Isolating the term in x.
x and y may be interchanged at this stage.
2
2x= +1
y−1 | DM1 | Condone  errors during simplification.
1 1
+
x−1 2 | A1 | OE
Allow ‘f−1’or ‘y =’ but NOT ‘x =’, nor fractions within
fractions.
[Domain of f−1 is] x1 | B1 | Accept — ∞ < x <1 or (— ∞, 1), condone [— ∞, 1).
4
Question | Answer | Marks | Guidance
--- 5(b) ---
5(b) | 1
gf  =−7
4 | B1
2x+1
= −7
2x−1 | M1 | 2x+1 1
Equating to their gf  .
2x−1 4
3
x=
8 | A1 | OE
Alternative solution for Question 5(b)
1
gf  =−7
4 | B1
x= f −1(−7) | M1 |  1
x= f −1 their gf  
 4
3
x=
8 | A1 | OE
3
Question | Answer | Marks | Guidance
The function f is defined by $\mathrm{f}(x) = \frac{2x + 1}{2x - 1}$ for $x < \frac{1}{2}$.

\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item State the value of f$(-1)$. [1]

\item \includegraphics{figure_5}

The diagram shows the graph of $y = \mathrm{f}(x)$. Sketch the graph of $y = \mathrm{f}^{-1}(x)$ on this diagram. Show any relevant mirror line. [2]

\item Find an expression for $\mathrm{f}^{-1}(x)$ and state the domain of the function $\mathrm{f}^{-1}$. [4]
\end{enumerate}
\end{enumerate}

The function g is defined by $\mathrm{g}(x) = 3x + 2$ for $x \in \mathbb{R}$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Solve the equation $\mathrm{f}(x) = \mathrm{gf}\left(\frac{1}{4}\right)$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2024 Q5 [10]}}
This paper (10 questions)
View full paper
Q1 5 Q2 5 Q3 5 Q4 6 Q5 10 Q6 6 Q7 8 Q8 10 Q9 10 Q10 10