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<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto 1+ax-{{x}^{2}}$, $x\in \mathbb{R}$, $x\le k$</p>
<p>where $a$ is a positive constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain with a sketch why the largest value of $k$ for which ${{\mathrm{f}}^{-1}}$ exists is $\frac{a}{2}$.</div><div class="list-item-paragraph">With this value of $k$, find ${{\mathrm{f}}^{-1}}$ in a similar form.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">If $k=1$ and $a=2$, find the set of values of $x$ for $\mathrm{f }{{\mathrm{f}}^{-1}}\left( x \right)={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">The function $\mathrm{g}$ is defined by</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{g}:x\mapsto \left\{ \begin{matrix}
2+2x-{{x}^{2}},0\le x\le 1, \\
1\,\,\,\,\,,1&lt;x&lt;2. \\
\end{matrix} \right.$</div><div class="list-item-paragraph">It is also given that $\mathrm{g}\left( x \right)=\mathrm{g}\left( x+2 \right)$ for all real values of $x$.</div><div class="list-item-paragraph">Sketch $y=\mathrm{g}\left( x \right)$ for $0\le x\le 5$.</div><p style="text-align: right;">[2]</p></li></ol>

$\mathrm{f}:x\mapsto 1+ax-{{x}^{2}}$, $x\in \mathbb{R}$, $x\le k$

2022 ASRJC J2 MYCT P1 Q8

Functions

<p>The function $\mathrm{f}$ is defined as follows.<br>\[\mathrm{f}:x\mapsto 1+4x-x^2,\quad x&lt;a.\]</p>
<p>You are given that&nbsp;$\mathrm{f}^{-1}$ exists.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-3-item-0" data-list-item-path="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-3-item-0"><div class="list-item-paragraph">By the method of completing the square, state the largest possible value of $a$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For the rest of the question, take $a=0$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-5-item-0" data-list-item-path="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-5-item-0"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{f}^{-1}(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-5-item-1" data-list-item-path="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-5-item-1"><div class="list-item-paragraph">Find the exact solution of the equation $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$.<span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined as follows.
\[\mathrm{f}:x\mapsto 1+4x-x^2,\quad x<a.\]

You are given that $\mathrm{f}^{-1}$ exists.

By the method of completing the square, state the largest possible value of $a$.

For the rest of the question, take $a=0$.

Find an expression for $\mathrm{f}^{-1}(x)$.

Find the exact solution of the equation $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$.

Inverse Functions

Horizontal Line Test

Self-inverse Functions: $ \mathrm{f} = \mathrm{f}^{-1}$

2025 RVHS Promo Q2

<p>Find the exact range of each of the following functions:</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \frac{x-1}{x+1},x\in \mathbb{R}, x &lt; 0$</p>

Find the exact range of each of the following functions:

$\mathrm{g}:x\mapsto \frac{x-1}{x+1},x\in \mathbb{R}, x < 0$

<p>The functions $\mathrm{f}$, $\mathrm{g}$, $\mathrm{h}$ and $\mathrm{k}$ are defined by</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $\mathrm{f}:x\,\mapsto \,{{\mathrm{e}}^{-x}}-\frac{1}{5}$, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $x\in \mathbb{R}$; <br></br></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $\mathrm{g}:x\,\mapsto \,\sqrt{\mathrm{f}\left( x \right)}$,  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $x\in \mathbb{R}$, $x\le \frac{3}{2}$; <br></br></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $\mathrm{h}:x\,\mapsto \,1-{{x}^{2}}$,  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $x\in \mathbb{R}$, $x\ge 0$; <br></br></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $\mathrm{k}:x\,\mapsto \,\frac{2}{x}$,  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $x\in \mathbb{R}$, $x&gt;0$. <br></br></p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $\mathrm{f}$, showing clearly any points of intersection with the axes and the equations of any asymptotes. Hence find the range of $\mathrm{f}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Solve the equation ${{\mathrm{g}}^{-1}}\mathrm{g}\left( x \right)=\mathrm{g}{{\mathrm{g}}^{-1}}\left( x \right)$, giving your answer in exact form.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Solve the equation $\mathrm{h}\left( x \right)={{\mathrm{h}}^{-1}}\left( x \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the function $\mathrm{gh}$ exists. Hence find the exact range of $\mathrm{gh}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Define the function ${{\mathrm{k}}^{2}}$ in a similar form and hence, evaluate ${{\mathrm{k}}^{2013}}\left( 4 \right)$.</div><p style="text-align: right;">[3]</p></li></ol>

The functions $\mathrm{f}$, $\mathrm{g}$, $\mathrm{h}$ and $\mathrm{k}$ are defined by

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $\mathrm{f}:x\,\mapsto \,{{\mathrm{e}}^{-x}}-\frac{1}{5}$, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $x\in \mathbb{R}$; <br></br>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $\mathrm{g}:x\,\mapsto \,\sqrt{\mathrm{f}\left( x \right)}$,  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $x\in \mathbb{R}$, $x\le \frac{3}{2}$; <br></br>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $\mathrm{h}:x\,\mapsto \,1-{{x}^{2}}$,  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $x\in \mathbb{R}$, $x\ge 0$; <br></br>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $\mathrm{k}:x\,\mapsto \,\frac{2}{x}$,  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  $x\in \mathbb{R}$, $x&gt;0$. <br></br>

2013 HCI C2 BT2 P1 Q11 [Modified]

<p>The diagrams below show the graphs of $y=\left| \mathrm{f}\left( x \right) \right|$ and $y=\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)$.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760687277269-z04p1kuzfq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA2ODcyNzcyNjktejA0cDFrdXpmcS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjI1MjYxLCJleHAiOjE3ODk2MjU1NjF9.KYt_Pyb3JOxRLauuIDM5vTvnCi2KjxS_5lH8wSei9BE" width="400" class="aligncenter" /></figure>
<p>On separate diagrams, sketch the graphs of:</p>
<ol class="i-list"><li><div class="list-item-paragraph">$y=\left| \mathrm{f}\left( 2x \right) \right|+1$</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">$y=\frac{1}{\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)}$</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">$y=\mathrm{f}\left( x \right)$</div><p style="text-align: right;">[3]</p></li></ol>
<p>showing clearly, in each case, the intersection(s) with the axes, the coordinates of the turning point(s) and the equation(s) of the asymptotes.</p>

The diagrams below show the graphs of $y=\left| \mathrm{f}\left( x \right) \right|$ and $y=\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)$.

On separate diagrams, sketch the graphs of:

showing clearly, in each case, the intersection(s) with the axes, the coordinates of the turning point(s) and the equation(s) of the asymptotes.

Modulus Functions

2018 EJC P2 Q2

<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">The function $\mathrm{g}$ is given by</span></div><div class="list-item-paragraph" style="text-align: center">$\mathrm{g}:x\mapsto \ln \left( \frac{{{e}^{x}}+5}{{{e}^{x}}-1} \right)$, for $x\in \mathbb{R}$, $x>0$.</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find ${{\mathrm{g}}^{-1}}\left( x \right)$ and states its domain.</span></div><div class="list-item-paragraph">The function $\mathrm{h}$ is defined by</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{h}:x\mapsto 1+\sqrt{9-{{\left( x-2 \right)}^{2}}}$, for $x\in \mathbb{R}$, $-1\le x\le 5$.</div></li><li><div class="list-item-content"><span class="list-item-text">Find the exact solutions of ${{\mathrm{g}}^{2021}}\mathrm{h}\left( x \right)=\ln 2$, giving your answer in its simplest form.</span></div></li></ol></li></ol>

2021 NYJC P1 Q5

<p>The graph of $y=2\ln x-{{\left( \ln x \right)}^{2}}$ where $x>0$ is shown in the diagram below. The graph cuts the $x$-axis at $x=1$ and $x={{e}^{2}}$.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759231661230-9gv0f8yr0fm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkyMzE2NjEyMzAtOWd2MGY4eXIwZm0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNTI2MSwiZXhwIjoxNzg5NjI1NTYxfQ.ZxVbOJuR7YB0NZ_xWph2AYLnTA5nSzSiFZOty03uzgg" width="400" class="aligncenter" /></figure>
<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto 2\ln x-{{\left( \ln x \right)}^{2}},$$0 < x\le a,$</p>
<p>where $a$ is a constant.</p>
<p>Find the greatest possible value of $a$ for which ${{\mathrm{f}}^{-1}}$ exists, giving your answer in exact form. Hence, find ${{\mathrm{f}}^{-1}}$ in similar form.</p><p style="text-align: right;">[6]</p>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \frac{1}{3}{{e}^{x}}$, $x\in \mathbb{R}$ .</p>
<p>Find the exact range of the composite function $\mathrm{g}{{\mathrm{f}}^{-1}}$.</p><p style="text-align: right;">[2]</p>

The graph of $y=2\ln x-{{\left( \ln x \right)}^{2}}$ where $x>0$ is shown in the diagram below. The graph cuts the $x$-axis at $x=1$ and $x={{e}^{2}}$.

$\mathrm{f}:x\mapsto 2\ln x-{{\left( \ln x \right)}^{2}},$$0 < x\le a,$

Find the greatest possible value of $a$ for which ${{\mathrm{f}}^{-1}}$ exists, giving your answer in exact form. Hence, find ${{\mathrm{f}}^{-1}}$ in similar form.

$\mathrm{g}:x\mapsto \frac{1}{3}{{e}^{x}}$, $x\in \mathbb{R}$ .

Find the exact range of the composite function $\mathrm{g}{{\mathrm{f}}^{-1}}$.

2018 SAJC MYE BT2 P1 Q5

<ol class="abc-list"><li><div class="list-item-paragraph">The function $\mathrm{f}$ is defined by</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{f}:x\mapsto \frac{2x-1}{2x-2},\quad x &lt; 1$.</div><div class="list-item-paragraph">Define ${{\mathrm{f}}^{-1}}$ in a similar form. Hence or otherwise find ${{\mathrm{f}}^{2}}(x)$ and state the value of ${{\mathrm{f}}^{1965}}(0.5)$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">The function $\mathrm{g}$ is defined by</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{g}:x\mapsto \sqrt{4+2x},\quad x\in \mathbb{R},\ \ 0\le x &lt; 16.$</div><ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{g}(x)$. Your sketch should indicate the coordinates of the end-points.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">On the same diagram of part <strong>(i)</strong>, sketch the graph of $y={{\mathrm{g}}^{-1}}(x)$, showing the relationship between $\mathrm{g}$ and ${{\mathrm{g}}^{-1}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the values of $x$ satisfying the equation ${{\mathrm{g}}^{-1}}\mathrm{g}(x)=\mathrm{g}{{\mathrm{g}}^{-1}}(x)$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Problems involving $\mathrm{f} ^n$ where  $n$ is large

2015 NYJC P2 Q2

<p>The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto \frac{4{{x}^{2}}-7x-1}{x-2}$, $x\in \mathbb{R},\ x\ne 2$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, indicating its turning points and asymptotes.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">State the range of $\mathrm{f}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The function $\mathrm{g}$, with domain the set of positive integers, is given by</p>
<p style="text-align: center">${\mathrm{g}(n) = \begin{cases} 
n, & \mathrm{for } n \mathrm{ odd} \\
2\mathrm{g}\left(\frac{n}{2}\right) + 1, & \mathrm{for } n \mathrm{ even} 
\end{cases}}$</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find $\mathrm{g}\left( 2 \right)$ and $\mathrm{g}\left( 8 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Does $\mathrm{fg}$ exist? Justify your answer.</div><p style="text-align: right;">[2]</p></li></ol>

The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto \frac{4{{x}^{2}}-7x-1}{x-2}$, $x\in \mathbb{R},\ x\ne 2$.

The function $\mathrm{g}$, with domain the set of positive integers, is given by

${\mathrm{g}(n) = \begin{cases} 
n, & \mathrm{for } n \mathrm{ odd} \\
2\mathrm{g}\left(\frac{n}{2}\right) + 1, & \mathrm{for } n \mathrm{ even} 
\end{cases}}$

2022 EJC Promo Q10

<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;$\mathrm{f}$ is a continuous function, explain, with the aid of a sketch, why the value of\[\lim_{n \to \infty} \frac{2}{n} \left\{ \mathrm{f}\left(1 + \frac{2}{n}\right) + \mathrm{f}\left(1 + \frac{4}{n}\right) + \dots + \mathrm{f}\left(1 + \frac{2n}{n}\right) \right\}\]is $\displaystyle \displaystyle\int_1^3 \mathrm{f}(x) \, \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence evaluate $\lim\limits_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{2k}{n}\right) \frac{2}{n}$ exactly.<span class="inline-marks">[4]</span></div></li></ol>

Riemann Sums

2025 NYJC P2 Q3

Applications of Integration

<p>The greatest integer function denoted by $\left[ x \right]$, is a function that gives the largest integer which is less than or equal to $x$. For example, $\left[ 1.5 \right]=1$, $\left[ 3 \right]=3$ and $\left[ -5.7 \right]=-6$.</p>
<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \left[ x \right]-x$, $x\in \mathbb{R}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $\mathrm{f}$ for $-3\le x\le 3$. Find the range of $\mathrm{f}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Explain why ${{\mathrm{f}}^{-1}}$ does not exist.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The functions $\mathrm{g}$ and $\mathrm{h}$ are defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \left[ x \right]-x$, $1\le x < 2$,</p>
<p style="text-align: center">$\mathrm{h}:x\mapsto {{\mathrm{e}}^{x}}$, $x>0$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Define ${{\mathrm{g}}^{-1}}$ in a similar form.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Explain why $\mathrm{h}{{\mathrm{g}}^{-1}}$ exists. Find the exact range of $\mathrm{h}{{\mathrm{g}}^{-1}}$.</div><p style="text-align: right;">[3]</p></li></ol>

The greatest integer function denoted by $\left[ x \right]$, is a function that gives the largest integer which is less than or equal to $x$. For example, $\left[ 1.5 \right]=1$, $\left[ 3 \right]=3$ and $\left[ -5.7 \right]=-6$.

$\mathrm{f}:x\mapsto \left[ x \right]-x$, $x\in \mathbb{R}$.

The functions $\mathrm{g}$ and $\mathrm{h}$ are defined by

$\mathrm{g}:x\mapsto \left[ x \right]-x$, $1\le x < 2$,

$\mathrm{h}:x\mapsto {{\mathrm{e}}^{x}}$, $x>0$.

2022 CJC J2 MYE P2 Q3

<p>The function $\mathrm{f}$ is defined by\[\mathrm{f} : x \mapsto \frac{ax - 3}{5x - 4}, \text{ for}\hspace{0.5em} x \in \mathbb{R}, x \neq \frac{4}{5}, \text{ and}\hspace{0.5em} a \in \mathbb{R}\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the range of $\mathrm{f}$ in terms of $a$.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The function $\mathrm{f}$ is such that&nbsp;$\mathrm{f}(x) = \mathrm{f}^{-1}(x)$ for all $x$ in the domain of $\mathrm{f}$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the value of $a$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find $\mathrm{f}^{2025}(2)$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is defined by\[\mathrm{g} : x \mapsto \left(x - \frac{1}{5}\right)^2 + \frac{2}{5}, \text{ for}\hspace{0.5em} x \in \mathbb{R}, 0 &lt; x &lt; \frac{3}{5}\]</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the range of $\mathrm{fg}$.</span><span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined by\[\mathrm{f} : x \mapsto \frac{ax - 3}{5x - 4}, \text{ for } x \in \mathbb{R}, x \neq \frac{4}{5}, \text{ and } a \in \mathbb{R}\]

Find the range of $\mathrm{f}$ in terms of $a$.

The function $\mathrm{f}$ is such that $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$ for all $x$ in the domain of $\mathrm{f}$.

Another function $\mathrm{g}$ is defined by\[\mathrm{g} : x \mapsto \left(x - \frac{1}{5}\right)^2 + \frac{2}{5}, \text{ for } x \in \mathbb{R}, 0 < x < \frac{3}{5}\]

2025 HCI P1 Q8

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759839407575-eijpwa31a2.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk4Mzk0MDc1NzUtZWlqcHdhMzFhMi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjI1MjYxLCJleHAiOjE3ODk2MjU1NjF9.2_M_1BtTy7_tUJOkQgi96Zd8-iqJoOM1khXLFJXL6UE" width="400" class="aligncenter" /></figure>
<p>The diagram shows the curve with equation $y=\mathrm{f}\left( x \right)$, for $x\in \mathbb{R},x\ne -\frac{1}{2},x\ne 2$. The curve crosses the axes at $x=-2,x=4$ and $y=9$, and has turning points with coordinates $\left( -4,7 \right)$ and $\left( 0.75,7 \right)$. The curve has asymptotes with equations $x=-0.5,x=2,y=0$ and $y=5$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the curve $y=\frac{1}{\mathrm{f}\left( x \right)}$, labelling any axial intercepts and turning points, and the equations of any asymptotes.</div><p style="text-align: right;">[4]</p></li></ol>
<p>For the remainder of this question, the domain of $\mathrm{f}$is $\left[ k,2 \right)\cup \left( 2,\infty \right)$, where $k$ is a constant.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State the least value of $k$ for which the function ${{\mathrm{f}}^{-1}}$ exists.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \frac{2x+a}{x-b}$ for $x\in \mathbb{R},x>b$,</p>
<p>where $a,b\in {{\mathbb{R}}^{+}}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the range of $\mathrm{g}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find ${{\left( \mathrm{fg} \right)}^{-1}}\left( 0 \right)$. Leave your answer in terms of $a$ and $b$.</div><p style="text-align: right;">[3]</p></li></ol>

The diagram shows the curve with equation $y=\mathrm{f}\left( x \right)$, for $x\in \mathbb{R},x\ne -\frac{1}{2},x\ne 2$. The curve crosses the axes at $x=-2,x=4$ and $y=9$, and has turning points with coordinates $\left( -4,7 \right)$ and $\left( 0.75,7 \right)$. The curve has asymptotes with equations $x=-0.5,x=2,y=0$ and $y=5$.

For the remainder of this question, the domain of $\mathrm{f}$is $\left[ k,2 \right)\cup \left( 2,\infty \right)$, where $k$ is a constant.

$\mathrm{g}:x\mapsto \frac{2x+a}{x-b}$ for $x\in \mathbb{R},x>b$,

2020 NJC P1 Q9

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{1}{2}{{e}^{1-{{x}^{2}}}},\mathrm{ }x\in \mathbb{R},\mathrm{ }x\le 1$</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \sqrt{1-\ln x},\mathrm{ }x\in \mathbb{R},\mathrm{ }0 < x\le e$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\mathrm{gf}$ exists, and find the range of $\mathrm{gf}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Justify, with a reason, whether ${{\mathrm{f}}^{-1}}$ exists.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The domain of $\mathrm{f}$ is restricted to $\left( -\infty ,b \right]$ such that $b$ is the largest value for which the inverse function ${{\mathrm{f}}^{-1}}$ exists. State the value of $b$ and define ${{\mathrm{f}}^{-1}}$clearly.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The graph of $y=\mathrm{h}\left( x \right)$ is obtained by transforming the graph of $y=\mathrm{g}(x)$ in the following $2$ steps.</div><div class="list-item-paragraph" style="margin-left: 2rem">Step 1: Scale parallel to the $x$-axis by a factor of $2$.</div><div class="list-item-paragraph" style="margin-left: 2rem">Step 2: Reflect in the $x$-axis.</div><div class="list-item-paragraph">Define $\mathrm{h}$ in a similar form.</div><p style="text-align: right;">[3]</p></li></ol>

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$\mathrm{f}:x\mapsto \frac{1}{2}{{e}^{1-{{x}^{2}}}},\mathrm{ }x\in \mathbb{R},\mathrm{ }x\le 1$

$\mathrm{g}:x\mapsto \sqrt{1-\ln x},\mathrm{ }x\in \mathbb{R},\mathrm{ }0 < x\le e$.

2016 HCI P1 Q11

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The function $\mathrm{f} $is defined as follows:</span></div><div class="list-item-paragraph" style="text-align: center;">$\mathrm{f}:x\mapsto 3\cos x-2\sin x,x\in \mathbb{R},-\pi \le x\le \pi $.</div><div class="list-item-paragraph">Write $\mathrm{f}\left( x \right)$ as $R\cos \,\left( x+\alpha \right)$, where $R$ and $\alpha $ are constants to be found. Hence, or otherwise, find the range of $\mathrm{f}$ and sketch the curve.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">The function $\mathrm{g}$ is defined as follows:</span></div><div class="list-item-paragraph" style="text-align: center;">$\mathrm{g}:x\mapsto 3\cos x-2\sin x,x\in \mathbb{R},-\alpha \le x\le b$.</div><div class="list-item-paragraph">Given that the function ${{\mathrm{g}}^{-1}}$ exists, write down the largest value of $b$. Find ${{\mathrm{g}}^{-1}}\left( x \right)$.</div><p style="text-align: right;">[3]</p></li></ol>

The function $\mathrm{f} $is defined as follows:

The function $\mathrm{g}$ is defined as follows:

2017 Specimen Paper P2 Q1

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \left| {{\mathrm{e}}^{x-3}}-1 \right|$, $x\le 3$</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto {{a}^{2}}-{{x}^{2}}$, $x\in \mathbb{R}$, $x&lt;0$, where&nbsp;$0 &lt; a &lt; 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760352543611-ovmvzctpy" data-list-item-path="item-1760352543611-ovmvzctpy"><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}$ in similar form.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760352544218-fzhmo4o0k" data-list-item-path="item-1760352544218-fzhmo4o0k"><div class="list-item-paragraph">Sketch the graphs of $y=\mathrm{g}\left( x \right)$, $y={{\mathrm{g}}^{-1}}\left( x \right)$ and $y={{\mathrm{g}}^{-1}}\mathrm{g}\left( x \right)$ on the same diagram.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760352545440-aru5mgx1i" data-list-item-path="item-1760352545440-aru5mgx1i"><div class="list-item-paragraph">Find the exact solution of $\mathrm{g}\left( x \right)={{\mathrm{g}}^{-1}}\left( x \right)$, leaving your answer in terms of $a$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760352546490-uqa8o2ayp" data-list-item-path="item-1760352546490-uqa8o2ayp"><div class="list-item-paragraph">Show that the composite function $\mathrm{fg}$ exists and find the range of $\mathrm{fg}$, leaving your answer in terms of $a$.<span class="inline-marks">[2]</span></div></li></ol>

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:

$\mathrm{f}:x\mapsto \left| {{\mathrm{e}}^{x-3}}-1 \right|$, $x\le 3$

$\mathrm{g}:x\mapsto {{a}^{2}}-{{x}^{2}}$, $x\in \mathbb{R}$, $x<0$, where $0 < a < 1$.

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