A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The function $\mathrm{f}$ and $\mathrm{g}$ are defined as</p>
<p style="margin-left: 11rem">$\mathrm{f}: x \mapsto \pi x, \quad x \in \mathbb{R}.$<br>$\mathrm{g}: x \mapsto \begin{cases} \quad \sin x & \text{for}\hspace{0.5em} 0 \leq x \leq \frac{\pi}{2}, \\ \cos x + 1 & \text{for}\hspace{0.5em} \frac{\pi}{2} &lt; x \leq \pi. \end{cases}$</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find $\mathrm{f}^{25}(x)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{g}(x)$ for&nbsp;$0 \le x \le \pi$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-3-item-2" data-list-item-path="imported-item-block-3-item-2"><div class="list-item-paragraph">If the domain of $\mathrm{g}$ is restricted to $D = (a, \pi]$ such that&nbsp;$\mathrm{g}^{-1}$ exists, state the smallest exact value of $a$. Hence find $\mathrm{g}^{-1}(x)$ and state the domain of $\mathrm{g}^{-1}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-3" data-list-item-path="imported-item-block-3-item-3"><div class="list-item-paragraph"><p>For the domain $D$ found in part <strong>(c)</strong>, show that the composite function $\mathrm{fg}$ exists and find the exact range of $\mathrm{fg}$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-4" data-list-item-path="imported-item-block-3-item-4"><div class="list-item-paragraph">Without finding $\mathrm{f}^{-1}$ and $\mathrm{fg}$, find the exact solution of $\mathrm{f}^{-1}(x) = \mathrm{g}\left(\frac{3\pi}{4}\right)$.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ and $\mathrm{g}$ are defined as

$\mathrm{f}: x \mapsto \pi x, \quad x \in \mathbb{R}.$
$\mathrm{g}: x \mapsto \begin{cases} \quad \sin x & \text{for } 0 \leq x \leq \frac{\pi}{2}, \\ \cos x + 1 & \text{for } \frac{\pi}{2} < x \leq \pi. \end{cases}$

Sketch the graph of $y = \mathrm{g}(x)$ for $0 \le x \le \pi$.

If the domain of $\mathrm{g}$ is restricted to $D = (a, \pi]$ such that $\mathrm{g}^{-1}$ exists, state the smallest exact value of $a$. Hence find $\mathrm{g}^{-1}(x)$ and state the domain of $\mathrm{g}^{-1}$.

Without finding $\mathrm{f}^{-1}$ and $\mathrm{fg}$, find the exact solution of $\mathrm{f}^{-1}(x) = \mathrm{g}\left(\frac{3\pi}{4}\right)$.

Existence of Composite Functions

Piecewise-Defined Functions

Inverse Trigonometric Functions

Restricting a Domain to Obtain an Inverse

Range of a Composite Function

Iterates $f^n$ for Large $n$

Inverse Functions

2025 HCI Promo Q8

Functions

<p>It is given that</p>
<p style="text-align: center">\[\mathrm{f}(x) = \begin{cases} \tan \left( \frac{\pi}{4a} x \right) & \text{for}\hspace{0.5em} -a &lt; x \le a, \\ \sin \left( \frac{\pi}{2a} x \right) & \text{for}\hspace{0.5em} a &lt; x \le 3a, \end{cases}\]</p>
<p>and that&nbsp;$\mathrm{f}(x) = \mathrm{f}(x - 4a)$ for all real values of $x$, where&nbsp;$a$ is a positive real constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{f}(x)$ for&nbsp;$-4a \le x \le 7a$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph"><p>By using the sketch in part <strong>(a)</strong>, write down an equation relating $\mathrm{f}(x)$ and $\mathrm{f}(-x)$.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-3-item-2" data-list-item-path="imported-item-block-3-item-2"><div class="list-item-paragraph">Show that&nbsp;$\int_{-2a}^{4a} \mathrm{f}(x) \,\mathrm{d}x = \frac{ka}{\pi} (1 + \ln m)$, where&nbsp;$k$ and $m$ are constants to be determined.<span class="inline-marks">[5]</span></div></li></ol>

\[\mathrm{f}(x) = \begin{cases} \tan \left( \frac{\pi}{4a} x \right) & \text{for } -a < x \le a, \\ \sin \left( \frac{\pi}{2a} x \right) & \text{for } a < x \le 3a, \end{cases}\]

and that $\mathrm{f}(x) = \mathrm{f}(x - 4a)$ for all real values of $x$, where $a$ is a positive real constant.

Sketch the graph of $y = \mathrm{f}(x)$ for $-4a \le x \le 7a$.

Show that $\int_{-2a}^{4a} \mathrm{f}(x) \,\mathrm{d}x = \frac{ka}{\pi} (1 + \ln m)$, where $k$ and $m$ are constants to be determined.

Integrable Forms of Trigonometric Functions

Periodic Functions

Odd and Even Functions

2026 HCI P1 Q7 [Modified]

Techniques of Integration

<p>It is given that</p>
<p style="text-align: center">\[\mathrm{f}(x) = \begin{cases} \quad\;\;\; \frac{1}{2}x^2, & \text{for}\hspace{0.5em} 1 \le x &lt; 2, \\ \frac{1}{4}(-3x + 14), & \text{for}\hspace{0.5em} 2 \le x &lt; 4, \end{cases}\]</p>
<p>and that&nbsp;$\mathrm{f}(x) = \mathrm{f}(x + 3)$ for all real values of $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">State the value of $\mathrm{f}(0)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{f}(x)$ for&nbsp;$0 \le x \le 5$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The function $\mathrm{g}$ is given by $\mathrm{g}(x) = \frac{4}{3}|x - 2|$ for&nbsp;$x \in \mathbb{R}$. By sketching the graph of $y = \mathrm{g}(x)$ on the same diagram as in <strong>part (b)</strong>, solve the inequality $\mathrm{f}(x) &gt; \mathrm{g}(x)$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">The function $\mathrm{h}$ is given by $\mathrm{h}(x) = \sqrt{3} \sin(\pi x) + \cos(\pi x) + 1$ for&nbsp;$0 \le x \le 2$. Explain why the composite function $\mathrm{hf}$ exists and find its range in exact form.<span class="inline-marks">[4]</span></div></li></ol>

\[\mathrm{f}(x) = \begin{cases} \quad\;\;\; \frac{1}{2}x^2, & \text{for } 1 \le x < 2, \\ \frac{1}{4}(-3x + 14), & \text{for } 2 \le x < 4, \end{cases}\]

and that $\mathrm{f}(x) = \mathrm{f}(x + 3)$ for all real values of $x$.

Modulus Functions

Graphical Method

Inequalities involving Modulus Functions

2024 RI Promo Q10

Inequalities

<p>It is given that&nbsp;$\mathrm{f}(x) = 6kx + x^2$, where&nbsp;$k &gt; 0$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = \mathrm{f}(x)$, indicating clearly the coordinates of the points where the curve crosses the axes and of the turning points, if any.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe the transformation that maps the graph of $y = \mathrm{f}(x)$ onto the graph of $y = x^2 - 9k^2$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that $\mathrm{g}(x) = \begin{cases} 6kx + x^2 & \text{for}\hspace{0.5em} -3k \leq x &lt; k \\ \quad 7k^2 & \text{for}\hspace{0.5em} k \leq x &lt; 2k \end{cases}$  $\quad$where $k &gt; 0$</p>
<p>and that&nbsp;$\mathrm{g}(x) = \mathrm{g}(x + 5k)$ for all real values of $x$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = \mathrm{g}(x)$ for&nbsp;$-3k \leq x &lt; 5k$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = \frac{1}{\mathrm{g}(x)}$ for&nbsp;$-3k \leq x &lt; 2k$.</span><span class="inline-marks">[3]</span></div></li></ol>

It is given that $\mathrm{f}(x) = 6kx + x^2$, where $k > 0$.

Sketch the graph of $y = \mathrm{f}(x)$, indicating clearly the coordinates of the points where the curve crosses the axes and of the turning points, if any.

Describe the transformation that maps the graph of $y = \mathrm{f}(x)$ onto the graph of $y = x^2 - 9k^2$.

It is given that $\mathrm{g}(x) = \begin{cases} 6kx + x^2 & \text{for } -3k \leq x < k \\ \quad 7k^2 & \text{for } k \leq x < 2k \end{cases}$  $\quad$where $k > 0$

and that $\mathrm{g}(x) = \mathrm{g}(x + 5k)$ for all real values of $x$.

Sketch the graph of $y = \mathrm{g}(x)$ for $-3k \leq x < 5k$.

Sketch the graph of $y = \frac{1}{\mathrm{g}(x)}$ for $-3k \leq x < 2k$.

2024 CJC P2 Q2

<p>The function $\mathrm{f}$ is given by</p>
<p>\[\mathrm{f}(x) = \begin{cases} 2 + \sqrt{2^2 - (x-2)^2}, & 0 &lt; x \leq 4, \\ 2 - \sqrt{2^2 - (x-6)^2}, & 4 &lt; x \leq 8. \end{cases}\]</p>
<p>It is given that&nbsp;$\mathrm{f}(x) = \mathrm{f}(x+8)$ for all real values of $x$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">On the diagram in the Printed Answer Book, sketch the graph of $y = \mathrm{f}(x)$ for&nbsp;$-6 \leq x \leq 7$, indicating clearly the coordinates of the end points and the points where the graph cuts the axes.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Without integrating, write down the exact area of the region bounded by $y = \mathrm{f}(x)$, the line $x = 4$, the $x$-axis and the $y$-axis.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The curve $C$ has equation $\frac{(y-3)^2}{a^2} - (x+2)^2 = 1$, where&nbsp;$a$ is a positive real constant.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">State the equations of the asymptotes of $C$ in terms of $a$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Determine the range of values of $a$ if there is at most one intersection between $C$ and the graph of $y = \mathrm{f}(x)$.</span><span class="inline-marks">[2]</span></div></li></ol>

\[\mathrm{f}(x) = \begin{cases} 2 + \sqrt{2^2 - (x-2)^2}, & 0 < x \leq 4, \\ 2 - \sqrt{2^2 - (x-6)^2}, & 4 < x \leq 8. \end{cases}\]

It is given that $\mathrm{f}(x) = \mathrm{f}(x+8)$ for all real values of $x$.

On the diagram in the Printed Answer Book, sketch the graph of $y = \mathrm{f}(x)$ for $-6 \leq x \leq 7$, indicating clearly the coordinates of the end points and the points where the graph cuts the axes.

Without integrating, write down the exact area of the region bounded by $y = \mathrm{f}(x)$, the line $x = 4$, the $x$-axis and the $y$-axis.

The curve $C$ has equation $\frac{(y-3)^2}{a^2} - (x+2)^2 = 1$, where $a$ is a positive real constant.

State the equations of the asymptotes of $C$ in terms of $a$.

Determine the range of values of $a$ if there is at most one intersection between $C$ and the graph of $y = \mathrm{f}(x)$.

Parameter Conditions for Graphs and Equations

Asymptotes and Discontinuities

Graphs of Conic Sections

2025 VJC P1 Q5

Graphing Techniques

Applications of Integration

<p>The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto 1 + \frac{3}{\mathrm{e}^x - 2}$ for&nbsp;$x \in \mathbb{R}$, $x &gt; \ln 2$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{f}^{-1}(x)$ and state its domain.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is defined by</p>
<p>\[\mathrm{g} : x \mapsto \begin{cases} 2(x - 1)^2 + 2 & \text{for}\hspace{0.5em} x \le 0, \\ 2 + |2 - x| & \text{for}\hspace{0.5em} 0 &lt; x \le 3. \end{cases}\]</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = \mathrm{g}(x)$ and explain why the composite function $\mathrm{f}^{-1}\mathrm{g}$ exists.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the exact value of $k$ such that&nbsp;$\mathrm{f}^{-1}\mathrm{g}(k) = \ln \frac{5}{2}$.</span><span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto 1 + \frac{3}{\mathrm{e}^x - 2}$ for $x \in \mathbb{R}$, $x > \ln 2$.

Find $\mathrm{f}^{-1}(x)$ and state its domain.

Another function $\mathrm{g}$ is defined by

\[\mathrm{g} : x \mapsto \begin{cases} 2(x - 1)^2 + 2 & \text{for } x \le 0, \\ 2 + |2 - x| & \text{for } 0 < x \le 3. \end{cases}\]

Sketch the graph of $y = \mathrm{g}(x)$ and explain why the composite function $\mathrm{f}^{-1}\mathrm{g}$ exists.

Hence find the exact value of $k$ such that $\mathrm{f}^{-1}\mathrm{g}(k) = \ln \frac{5}{2}$.

Composite Functions

2025 RI P1 Q7

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The curves $C_{1}$ and $C_{2}$ have equations $\frac{x^2}{9}-\frac{y^2}{4}=1$ and $y^2+x^2=k^2$ respectively, where&nbsp;$k$ is a positive constant.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch $C_{1}$ and $C_{2}$ on the same diagram, stating the coordinates of any points of intersection with the axes and the equations of any asymptotes.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">State the range of values of $k$ for&nbsp;$C_{1}$ and $C_{2}$ to intersect.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">State the equations of the common lines of symmetry for both&nbsp;$C_{1}$ and $C_{2}$.</span><span class="inline-marks">[1]</span></div></li></ol></li><li><div class="list-item-paragraph">The function $\mathrm{f}$, with domain the set of all real values, is given by\[\mathrm{f}(x)=\begin{cases}-2x+6 & \text{for}\hspace{0.5em} 0 &lt; x \le 3, \\ 3x-9 & \text{for}\hspace{0.5em} 3 &lt; x \le 5, \end{cases}\]and that&nbsp;$\mathrm{f}(x)=\mathrm{f}(x+5)$.</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{f}(46)$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y=\mathrm{f}(x)$ for&nbsp;$-5 \le x \le 5$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, state the roots of $\mathrm{f}(-x)=0$ for&nbsp;$-5 \le x \le 5$.</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

The curves $C_{1}$ and $C_{2}$ have equations $\frac{x^2}{9}-\frac{y^2}{4}=1$ and $y^2+x^2=k^2$ respectively, where $k$ is a positive constant.

Circle–Curve Intersection Conditions

2025 JPJC P1 Q5

<p>It is given that</p>
<p>\[\mathrm{f}(x) = \left\{ 
\begin{array}{ll}
4-2x & \text{for}\hspace{0.5em} 0 &lt; x  \le 2 \\
-(x - 4)^2 + 4 & \text{for}\hspace{0.5em} 2 &lt; x  \le 4
\end{array} 
\right.\]</p>
<p>and $\mathrm{f}(x) = \mathrm{f}(x + 4)$ for all real values of $x$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Evaluate $ \mathrm{f}(17) + \mathrm{f}(-8)$.</div></li><li><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{f}(x)$ for&nbsp;$-10 \le x \le 7$.</div></li></ol>

\[\mathrm{f}(x) = \left\{ 
\begin{array}{ll}
4-2x & \text{for } 0 < x  \le 2 \\
-(x - 4)^2 + 4 & \text{for } 2 < x  \le 4
\end{array} 
\right.\]

and $\mathrm{f}(x) = \mathrm{f}(x + 4)$ for all real values of $x$.

Worked Example 4

<p>Sketch the graph of $\mathrm{f}\left( x \right)=\left\{ \begin{matrix}-1 \\x \\1\mathrm{ } \\\end{matrix}\right.\mathrm{}\begin{matrix}\mathrm{for} \\\mathrm{for}\\\mathrm{for}\\\end{matrix}\mathrm{   }\begin{matrix}x<-1 \\ -1\le x\le 1 \\x>1  \\\end{matrix}$ and state its range

</p>
<p>Determine whether $\mathrm{f}$ is a one-one function.</p>

Sketch the graph of $\mathrm{f}\left( x \right)=\left\{ \begin{matrix}-1 \\x \\1\mathrm{ } \\\end{matrix}\right.\mathrm{}\begin{matrix}\mathrm{for} \\\mathrm{for}\\\mathrm{for}\\\end{matrix}\mathrm{   }\begin{matrix}x<-1 \\ -1\le x\le 1 \\x>1  \\\end{matrix}$ and state its range



Determine whether $\mathrm{f}$ is a one-one function.

One-to-One Functions and Inverse Existence

Range of a Function

Worked Example 3

<p>It is given that \[\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
   4-2x\text{               for   }0&lt;x\le 2,  \\
   -{{\left( x-4 \right)}^{2}}+4\text{    for   }2&lt;x\le 4,  \\
\end{matrix} \right.\]<br></p>
<p>and $\mathrm{f}(x) = f(x + 4)$ for all real values of $x$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Evaluate $\mathrm{f}\left( 17 \right)+\mathrm{f}\left( -8 \right)$.</div></li><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for&nbsp;$-10\le x\le 7$.</div></li></ol>

It is given that \[\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
   4-2x\text{               for   }0<x\le 2,  \\
   -{{\left( x-4 \right)}^{2}}+4\text{    for   }2<x\le 4,  \\
\end{matrix} \right.\]


and $\mathrm{f}(x) = f(x + 4)$ for all real values of $x$.

<p>Sketch the graph of $\mathrm{f}\left( x \right)=\left\{ \begin{matrix}-1 \\x \\1\mathrm{ } \\\end{matrix}\right.\mathrm{}\begin{matrix}\mathrm{for} \\\mathrm{for}\\\mathrm{for}\\\end{matrix}\mathrm{   }\begin{matrix}x&lt;-1 \\ -1\le x\le 1 \\x&gt;1  \\\end{matrix}$ and state its range</p>
<p>Determine whether $f$ is a one-one function.</p>

Sketch the graph of $\mathrm{f}\left( x \right)=\left\{ \begin{matrix}-1 \\x \\1\mathrm{ } \\\end{matrix}\right.\mathrm{}\begin{matrix}\mathrm{for} \\\mathrm{for}\\\mathrm{for}\\\end{matrix}\mathrm{   }\begin{matrix}x<-1 \\ -1\le x\le 1 \\x>1  \\\end{matrix}$ and state its range

Determine whether $f$ is a one-one function.

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The function $\mathrm{f}$ is defined by</div><div class="list-item-paragraph">\[\mathrm{f}(x) = \left\{ 
\begin{array}{ll}
2x + 1 & \text{for}\hspace{0.5em} 0 \le x &lt; 2 \\
(x - 4)^2 + 1 & \text{for}\hspace{0.5em} 2 \le x &lt; 4.
\end{array} 
\right.\]</div><div class="list-item-paragraph">It is further given that&nbsp;$\mathrm{f}\left( x \right)=\mathrm{f}\left( x+4 \right)$ for all real values of $x$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the values of $\mathrm{f}\left( 1 \right)$ and $\mathrm{f}\left( 5 \right)$. Hence, explain why $\mathrm{f}$ is not one-one.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for&nbsp;$-4\le x&lt;8$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the range of $\mathrm{f}$ for&nbsp;$-4\le x&lt;8$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

2014 SRJC Promo Q11

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">${\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
   2-\ln \left( x+\mathrm{e} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for }\,\,\,\,\,\,-\mathrm{e} &lt; x &lt; 0  \\
   \cos \left( \frac{\pi }{2}x \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for }\,\,\,\,\,\,\,\,0\le x\le 1  \\
\end{matrix} \right.}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760869375877-bf343a0g4" data-list-item-path="item-1760869375877-bf343a0g4"><div class="list-item-paragraph">Given that the inverse function ${{\mathrm{f}}^{-1}}$ exists, find the exact value of $m$ if&nbsp;${{\mathrm{f}}^{-1}}\left( m \right)=-1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}\left( x \right)=\sin x$, $\qquad 0\le x\le \frac{\pi }{4}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760869400483-1t29ygu4r" data-list-item-path="item-1760869400483-1t29ygu4r"><div class="list-item-paragraph">Explain why the composite function $\mathrm{fg}$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760869401219-r6rxts5i9" data-list-item-path="item-1760869401219-r6rxts5i9"><div class="list-item-paragraph">Find the exact value of $\mathrm{fg}\left( \frac{\pi }{6} \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The domain of $\mathrm{f}$ is further restricted to $0\le x\le 1$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-1760869426224-3w0amd661" data-list-item-path="item-1760869426224-3w0amd661"><div class="list-item-paragraph">By considering ${{\mathrm{f}}^{2}}\left( 1 \right)$ and ${{\mathrm{f}}^{3}}\left( 1 \right)$, deduce the value(s) of ${{\mathrm{f}}^{n}}\left( 1 \right)$ where&nbsp;$n$ is a positive integer.<span class="inline-marks">[2]</span></div></li></ol>

${\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
   2-\ln \left( x+\mathrm{e} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for }\,\,\,\,\,\,-\mathrm{e} < x < 0  \\
   \cos \left( \frac{\pi }{2}x \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for }\,\,\,\,\,\,\,\,0\le x\le 1  \\
\end{matrix} \right.}$.

Another function $\mathrm{g}$ is defined by

$\mathrm{g}\left( x \right)=\sin x$, $\qquad 0\le x\le \frac{\pi }{4}$.

The domain of $\mathrm{f}$ is further restricted to $0\le x\le 1$.

2024 HCI Promo Q8

<p>A function $g$ is said to be self-inverse if&nbsp;${{\mathrm{g}}^{-1}}\left( x \right)=\mathrm{g}\left( x \right)$ for all $x$ in the domain of $g$.</p>
<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}\left( x \right)=\frac{ax+b}{x-2}$, $x\in \mathbb{R}$, $x\ne 2$,</p>
<p>where&nbsp;$a$ and $b$ are real constants.</p>
<p>It is given that&nbsp;$\mathrm{f}$ is self-inverse.</p>
<ol class="i-list"><li data-list-item-id="item-1760754661230-wjvgxti4y" data-list-item-path="item-1760754661230-wjvgxti4y"><div class="list-item-paragraph">Find $a$ and the value(s) that&nbsp;$b$ cannot take.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760754662182-gu7tautxi" data-list-item-path="item-1760754662182-gu7tautxi"><div class="list-item-paragraph">Explain why the composite function ${{\mathrm{f}}^{n}}$ always exists for all integers $n &gt; 1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760754663285-l3fjvzny4" data-list-item-path="item-1760754663285-l3fjvzny4"><div class="list-item-paragraph">Hence find ${{\mathrm{f}}^{2025}}\left( 1 \right)$ in terms of $b$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The function $\mathrm{h}$ is defined for all $x\in \mathbb{R}$ by</p>
<p style="text-align: center">$\mathrm{h}\left( x \right)=\left\{ \begin{matrix}
   3x+2\,\,\,\,\,\,\,\,x\ge 0,  \\
   2{{x}^{2}}\,\,\,\,\,\,\,\,\,\,\,\,\,x&lt;0.  \\
\end{matrix} \right.$</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760754729585-ngk7kpfvc" data-list-item-path="item-1760754729585-ngk7kpfvc"><div class="list-item-paragraph">Sketch the graph of $\mathrm{h}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760754730545-uain7zreu" data-list-item-path="item-1760754730545-uain7zreu"><div class="list-item-paragraph">Using the value of $a$ found in <strong>(i)</strong> and given that&nbsp;$b=1$, find $\mathrm{hf}\left( \frac{1}{2} \right)$.<span class="inline-marks">[1]</span></div></li></ol>

A function $g$ is said to be self-inverse if ${{\mathrm{g}}^{-1}}\left( x \right)=\mathrm{g}\left( x \right)$ for all $x$ in the domain of $g$.

$\mathrm{f}\left( x \right)=\frac{ax+b}{x-2}$, $x\in \mathbb{R}$, $x\ne 2$,

It is given that $\mathrm{f}$ is self-inverse.

The function $\mathrm{h}$ is defined for all $x\in \mathbb{R}$ by

$\mathrm{h}\left( x \right)=\left\{ \begin{matrix}
   3x+2\,\,\,\,\,\,\,\,x\ge 0,  \\
   2{{x}^{2}}\,\,\,\,\,\,\,\,\,\,\,\,\,x<0.  \\
\end{matrix} \right.$

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

2024 SAJC P2 Q3

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
   2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\mathrm{ }1\le x&lt;8,  \\
   \frac{x}{2}+12\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\mathrm{ }8\le x\le 30.  \\
\end{matrix} \right.$</p>
<ol class="abc-list"><li data-list-item-id="item-1760754423429-3tkk864r0" data-list-item-path="item-1760754423429-3tkk864r0"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760754423969-rud5gmiwy" data-list-item-path="item-1760754423969-rud5gmiwy"><div class="list-item-paragraph">Solve the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760754424967-bas5460nr" data-list-item-path="item-1760754424967-bas5460nr"><div class="list-item-paragraph">Show that the composite function ${{\mathrm{f}}^{2}}$ exists and find its range.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760754426572-7egptocfi" data-list-item-path="item-1760754426572-7egptocfi"><div class="list-item-paragraph">Given that the composite function ${{\mathrm{f}}^{n}}$ exists for&nbsp;$n\ge 3$, find the range of ${{\mathrm{f}}^{3}}$, ${{\mathrm{f}}^{4}}$ and hence find the range of ${{\mathrm{f}}^{n}}$ as&nbsp;$n\to \infty $.<span class="inline-marks">[3]</span></div></li></ol>

$\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
   2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\mathrm{ }1\le x<8,  \\
   \frac{x}{2}+12\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\mathrm{ }8\le x\le 30.  \\
\end{matrix} \right.$

Intersections of a Function and Its Inverse

2024 DHS P2 Q4

<ol class="abc-list"><li data-list-item-id="item-1760695372830-3qcqpg4zl" data-list-item-path="item-1760695372830-3qcqpg4zl"><div class="list-item-paragraph">The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</div><div class="list-item-paragraph" style="text-align: center">${\mathrm{f}:x\rightarrow\frac{x-2}{x+2},\quad x\in\mathbb{R},\; x\neq -2,}$</div><div class="list-item-paragraph" style="text-align: center">${\mathrm{g}:x\rightarrow -x^2,\quad x\in\mathbb{R},\; x &lt; -\sqrt{2},}$</div><ol class="i-list"><li data-list-item-id="subitem-1760695392997-5fvml0sqs" data-list-item-path="item-1760695372830-3qcqpg4zl/subitem-1760695392997-5fvml0sqs"><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state its domain.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760695402716-fx2neqsjs" data-list-item-path="item-1760695372830-3qcqpg4zl/subitem-1760695402716-fx2neqsjs"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{fg}\left( x \right)$ and hence, or otherwise, find ${{\left( \mathrm{fg} \right)}^{-1}}\left( 3 \right)$.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="item-1760695373526-07tjvnbaj" data-list-item-path="item-1760695373526-07tjvnbaj"><div class="list-item-paragraph">It is given that</div><div class="list-item-paragraph" style="text-align: center">${\mathrm{h}(x)=\begin{cases} -4x-12 & \text{for}\hspace{0.5em} -4 \leq x \leq -2, \\ x\left|x\right| & \text{for}\hspace{0.5em} -2 &lt; x \leq -2 \end{cases}}$</div><div class="list-item-paragraph">and that&nbsp;$\mathrm{h}\left( x \right)=\mathrm{h}\left( x+6 \right)$ for all real values of $x$.</div><ol class="i-list"><li data-list-item-id="subitem-1760695466481-iguzm1f62" data-list-item-path="item-1760695373526-07tjvnbaj/subitem-1760695466481-iguzm1f62"><div class="list-item-paragraph">Evaluate $\mathrm{h}\left( -4 \right)$ and $\mathrm{h}\left( 12 \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760695467066-0lyti35da" data-list-item-path="item-1760695373526-07tjvnbaj/subitem-1760695467066-0lyti35da"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{h}\left( x \right)$ for&nbsp;$-4\le x\le 6$ and explain why $\mathrm{h}$ has no inverse.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

2022 JPJC P1 Q9

<ol class="abc-list"><li data-list-item-id="item-1760695205137-jyw0y6jwm" data-list-item-path="item-1760695205137-jyw0y6jwm"><div class="list-item-paragraph">A curve $y=\mathrm{h}\left( x \right)$ intersects the $x$-axis at point $A$ with coordinates $\left( a,0 \right)$, where&nbsp;$a\ne 0$, and has a horizontal asymptote with equation $y=b$, where&nbsp;$b$ is a non-zero constant.</div><div class="list-item-paragraph">For each of the following curves, state, if it is possible to do so, the equation of the horizontal asymptote and the coordinates of the point where the curve intersects the $x$-axis.</div><ol class="i-list"><li data-list-item-id="subitem-1760695213304-52h33tiwo" data-list-item-path="item-1760695205137-jyw0y6jwm/subitem-1760695213304-52h33tiwo"><div class="list-item-paragraph">$y=2\mathrm{h}\left( x+1 \right)$</div></li><li data-list-item-id="subitem-1760695215276-15feu3mta" data-list-item-path="item-1760695205137-jyw0y6jwm/subitem-1760695215276-15feu3mta"><div class="list-item-paragraph">$y=\mathrm{h}\left( -x \right)$</div></li><li data-list-item-id="subitem-1760695215817-kdvhh0tfb" data-list-item-path="item-1760695205137-jyw0y6jwm/subitem-1760695215817-kdvhh0tfb"><div class="list-item-paragraph">$y=\frac{1}{\mathrm{h}\left( x \right)}$<span class="inline-marks">[5]</span></div></li></ol></li><li data-list-item-id="item-1760695205669-acmyl9iqt" data-list-item-path="item-1760695205669-acmyl9iqt"><div class="list-item-paragraph">It is given that</div><div class="list-item-paragraph" style="text-align: center"><p>${\mathrm{f}(x) = \begin{cases} \sqrt{x} &amp; \text{for}\hspace{0.5em} 0 \le x \le 1, \\ -\frac{1}{2}x + \frac{3}{2} &amp; \text{for}\hspace{0.5em} 1 &lt; x \le 3, \end{cases}}$</p></div><div class="list-item-paragraph">and that&nbsp;$\mathrm{f}\left( x \right)=\mathrm{f}\left( x+3 \right)$ for all real values of $x$.</div><ol class="i-list"><li data-list-item-id="subitem-1760695289444-gxvlkvsxp" data-list-item-path="item-1760695205669-acmyl9iqt/subitem-1760695289444-gxvlkvsxp"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for&nbsp;$-3\le x\le 5$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760695290165-u70too8cg" data-list-item-path="item-1760695205669-acmyl9iqt/subitem-1760695290165-u70too8cg"><div class="list-item-paragraph">Hence find the value of $\displaystyle\displaystyle \displaystyle\int_{-3}^{5}{\mathrm{f}\left( x \right)\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Composite Transformations

Area Bounded by Curves

Reciprocal Graph, $y=\frac{1}{f(x)}$

Translations, Reflections and Scalings of Graphs

2023 MI P1 Q9

Transformations of Graphs

<ol class="abc-list"><li data-list-item-id="item-1760688654297-w40ilgae2" data-list-item-path="item-1760688654297-w40ilgae2"><div class="list-item-paragraph">Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</div><div class="list-item-paragraph" style="text-align: center">${\mathrm{f}(x) = e^{x+3} - a, \; x \in \mathbb{R}, \; x \leq 4,}$<br>${\mathrm{g}(x) = 2x + 5 \text{ for}\hspace{0.5em} x \in \mathbb{R}.}$</div><div class="list-item-paragraph">where&nbsp;$a$ is a constant.</div><ol class="i-list"><li data-list-item-id="subitem-1760688676482-2dyabgj5y" data-list-item-path="item-1760688654297-w40ilgae2/subitem-1760688676482-2dyabgj5y"><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state its domain, leaving your answer in terms of $a$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760688677360-oaxa4ymog" data-list-item-path="item-1760688654297-w40ilgae2/subitem-1760688677360-oaxa4ymog"><div class="list-item-paragraph">It is given that the composite function ${{\mathrm{g}}^{-1}}\mathrm{f}$ exists and ${{\mathrm{g}}^{-1}}\mathrm{f}\left( -3 \right)=15$. Without finding ${{\mathrm{g}}^{-1}}\mathrm{f}$, find the value of $a$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1760688655145-ki53hjody" data-list-item-path="item-1760688655145-ki53hjody"><div class="list-item-paragraph">A curve has equation $y=\mathrm{h}\left( x \right)$, where</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{h}\left( x \right)\left\{ \begin{matrix}
{{2}^{x}}\,\,\,\mathrm{for}\,\,\,0&lt;x\le 2, \\
6-\frac{1}{2}{{x}^{2}}\,\,\,\mathrm{for}\,\,\,2&lt;x\le 3, \\
\end{matrix} \right.$</div><div class="list-item-paragraph">and that&nbsp;$\mathrm{h}\left( x \right)=\mathrm{h}\left( x-3 \right)$ for all real values of $x$.</div><ol class="i-list"><li data-list-item-id="subitem-1760688764563-pzzxz93gx" data-list-item-path="item-1760688655145-ki53hjody/subitem-1760688764563-pzzxz93gx"><div class="list-item-paragraph">Find $\mathrm{h}\left( 9 \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760688766640-s2f8xhhe7" data-list-item-path="item-1760688655145-ki53hjody/subitem-1760688766640-s2f8xhhe7"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{h}\left( x \right)$ for&nbsp;$-1\le x\le 4$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

2023 ACJC Promo Q5

<p>It is given that</p>
<p style="text-align: center">${\mathrm{f}:x\mapsto\begin{cases} (x-2)^2, & 0 &lt; x \leq 3, \\ 3x-8, & 3 &lt; x \leq 4, \end{cases}}$</p>
<ol class="abc-list"><li data-list-item-id="item-1760427803262-n9kbflu4x" data-list-item-path="item-1760427803262-n9kbflu4x"><ol class="i-list"><li data-list-item-id="subitem-1760427804042-d13ixls8o" data-list-item-path="item-1760427803262-n9kbflu4x/subitem-1760427804042-d13ixls8o"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, labelling the coordinates of any turning points and endpoints. Explain why ${{\mathrm{f}}^{-1}}$ does not exist.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760427806044-p2nbvvexg" data-list-item-path="item-1760427803262-n9kbflu4x/subitem-1760427806044-p2nbvvexg"><div class="list-item-paragraph">If the domain of $\mathrm{f}$ is restricted to $\left( 0,k \right]$, state the largest value of $k$ such that&nbsp;${{\mathrm{f}}^{-1}}$ exists. Hence, for this value of $k$, find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760427806607-nggifbk2x" data-list-item-path="item-1760427803262-n9kbflu4x/subitem-1760427806607-nggifbk2x"><div class="list-item-paragraph">The function $\mathrm{g}$ is such that</div><div class="list-item-paragraph" style="text-align: center">$g:x\mapsto {{\mathrm{e}}^{x}}+3$, $x\le 0$.</div><div class="list-item-paragraph">Find the function $\mathrm{fg}$, giving your answer in similar form.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760427847070-w2rk53g5z" data-list-item-path="item-1760427847070-w2rk53g5z"><div class="list-item-paragraph">It is given further that&nbsp;$\mathrm{f}\left( x \right)=\mathrm{f}\left( x+4 \right)$.</div><ol class="i-list"><li data-list-item-id="subitem-1760427849777-bstl6r9ya" data-list-item-path="item-1760427847070-w2rk53g5z/subitem-1760427849777-bstl6r9ya"><div class="list-item-paragraph">Evaluate $\mathrm{f}\left( 25 \right)$ and $\mathrm{f}\left( -8 \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760427850841-74tdhz4uo" data-list-item-path="item-1760427847070-w2rk53g5z/subitem-1760427850841-74tdhz4uo"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( \frac{1}{2}x-1 \right)$ for&nbsp;$-8 &lt; x\le 10$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

${\mathrm{f}:x\mapsto\begin{cases} (x-2)^2, & 0 < x \leq 3, \\ 3x-8, & 3 < x \leq 4, \end{cases}}$

2023 CJC P1 Q7

<p>It is given that $\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
\sqrt{x} &amp; \mathrm{for }\!\!~\!\!\mathrm{ }\,\,\,\,\,0\le x\le 1, \\
-\frac{1}{2}x+\frac{3}{2} &amp; \mathrm{for }\!\!~\!\!\mathrm{ }\!\!~\!\!\mathrm{ }\,\,\,\,\,1&lt;x\le 3, \\
\end{matrix} \right.$ and that $\mathrm{f}\left( x \right)=\mathrm{f}\left( x+3 \right)$ for all real values of $x$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for $-3\le x\le 5$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find the value of $\displaystyle\int_{-3}^{5}{\mathrm{f}\left( x \right)\,\mathrm{d}x}$.</div><p style="text-align: right;">[2]</p></li></ol>

It is given that $\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
\sqrt{x} &amp; \mathrm{for }\!\!~\!\!\mathrm{ }\,\,\,\,\,0\le x\le 1, \\
-\frac{1}{2}x+\frac{3}{2} &amp; \mathrm{for }\!\!~\!\!\mathrm{ }\!\!~\!\!\mathrm{ }\,\,\,\,\,1&lt;x\le 3, \\
\end{matrix} \right.$ and that $\mathrm{f}\left( x \right)=\mathrm{f}\left( x+3 \right)$ for all real values of $x$.

Math Question Bank