CAIE P1 2024 June — Question 6 7 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComposite & Inverse Functions
TypeFind inverse function
DifficultyModerate -0.3 This is a straightforward inverse function question with routine procedures: sketching using reflection in y=x, finding inverse by swapping and rearranging (simple algebraic manipulation with square root), solving a basic equation, and explaining why graphs don't intersect. All parts are standard textbook exercises requiring no novel insight, making it slightly easier than average.
Spec1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x)
\includegraphics{figure_6} The function f is defined by f\((x) = \frac{2}{x^2} + 4\) for \(x < 0\). The diagram shows the graph of \(y = \text{f}(x)\).
  1. On this diagram, sketch the graph of \(y = \text{f}^{-1}(x)\). Show any relevant mirror line. [2]
  2. Find an expression for f\(^{-1}(x)\). [3]
  3. Solve the equation f\((x) = 4.5\). [1]
  4. Explain why the equation f\(^{-1}(x) = \text{f}(x)\) has no solution. [1]
Question 6:
AnswerMarks Guidance
6(a)B1 For curve in correct quadrant.
B1Fully correct including line yx.
Horizontal asymptote closer to x axis than vertical
asymptote is to y axis.
2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
6(b)2
x  2  4 leading to y2x42 or y2  x4
AnswerMarks Guidance
y 2M1 Allow x and y swapped around.
2 2
y2  leading to y 2 or x
AnswerMarks
x4 x4 y4M1
2
f1x  
 
AnswerMarks
x4A1
3
AnswerMarks Guidance
6(c)x2 B1
1
AnswerMarks Guidance
6(d)Because f1 is always negative and f is always positive or curves do not
intersectB1 Accept other correct answers e.g. ‘f is only defined for
positive values of x and f–1 is only defined for negative
values of x’ or ‘domains do not overlap’ or ‘the y values
cannot be the same’ or ‘the x values cannot be the same’.
1
AnswerMarks Guidance
QuestionAnswer Marks 
Question 6:
--- 6(a) ---
6(a) | B1 | For curve in correct quadrant.
B1 | Fully correct including line yx.
Horizontal asymptote closer to x axis than vertical
asymptote is to y axis.
2
Question | Answer | Marks | Guidance
--- 6(b) ---
6(b) | 2
x  2  4 leading to y2x42 or y2  x4
y 2 | M1 | Allow x and y swapped around.
2 2
y2  leading to y 2 or x
x4 x4 y4 | M1
2
f1x  
 
x4 | A1
3
--- 6(c) ---
6(c) | x2 | B1
1
--- 6(d) ---
6(d) | Because f1 is always negative and f is always positive or curves do not
intersect | B1 | Accept other correct answers e.g. ‘f is only defined for
positive values of x and f–1 is only defined for negative
values of x’ or ‘domains do not overlap’ or ‘the y values
cannot be the same’ or ‘the x values cannot be the same’.
1
Question | Answer | Marks | Guidance
\includegraphics{figure_6}

The function f is defined by f$(x) = \frac{2}{x^2} + 4$ for $x < 0$. The diagram shows the graph of $y = \text{f}(x)$.

\begin{enumerate}[label=(\alph*)]
\item On this diagram, sketch the graph of $y = \text{f}^{-1}(x)$. Show any relevant mirror line. [2]

\item Find an expression for f$^{-1}(x)$. [3]

\item Solve the equation f$(x) = 4.5$. [1]

\item Explain why the equation f$^{-1}(x) = \text{f}(x)$ has no solution. [1]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2024 Q6 [7]}}
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