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>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.

Piecewise-Defined Functions

Integrable Forms of Trigonometric Functions

Periodic Functions

Odd and Even Functions

2026 HCI P1 Q7 [Modified]

Functions

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 $x$.

Existence of Composite Functions

Modulus Functions

Graphical Method

Range of a Composite Function

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

<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>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 $x$.

<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>

One-to-One Functions and Inverse Existence

Range of a Function

2014 SRJC Promo Q11

<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>

Composite Functions

Inverse Functions

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>

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

Restricting a Domain to Obtain an Inverse

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 $x$.

SAJC Functions Tutorial Q3

<p>It is given that</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \left\{ \begin{matrix}
-a\sqrt{1-\frac{{{x}^{2}}}{4},}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-2x\le x&lt;2 \\
3x-6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x\le x&lt;3 \\
\end{matrix} \right.$</p>
<p>and that $\mathrm{f}\left( x \right)=\mathrm{f}\left( x+5 \right)$ for all real values of $x$, where $a$ is a positive constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{f}\left( 2021 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for $-5\le x\le 7$, stating the coordinates of the maximum points, end-points and points where the curve crosses the $x$- and $y$-axes.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">On a separate diagram, sketch the curve with equation $y=3\mathrm{f}\left( x+2 \right)$ for $-2\le x\le 3$, stating the coordinates of the points where the curve crosses the $x$- and $y$-axes.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Another function $\mathrm{g}$ is defined by</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{g}:x\mapsto \sqrt{x}+2$, for $0\le x<1$.</div><div class="list-item-paragraph">State the range of $\mathrm{g}$ and find $\mathrm{fg}\left( x \right)$, stating its domain.</div><p style="text-align: right;">[3]</p></li></ol>

$\mathrm{f}:x\mapsto \left\{ \begin{matrix}
-a\sqrt{1-\frac{{{x}^{2}}}{4},}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-2x\le x&lt;2 \\
3x-6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x\le x&lt;3 \\
\end{matrix} \right.$

and that $\mathrm{f}\left( x \right)=\mathrm{f}\left( x+5 \right)$ for all real values of $x$, where $a$ is a positive constant.

2021 JPJC P1 Q6

<p>It is given that</p>
<p style="text-align: center">$\mathrm{f}\left( x \right)={{x}^{2}}-6x+5$, for $1 < x\le 5$,</p>
<p style="text-align: center">$\mathrm{g}\left( x \right)={{x}^{2}}-6x+5$, for $1 < x < 3$,</p>
<p style="text-align: center">$\mathrm{h}\left( x \right)=\left\{ \begin{matrix}
\ln x\,\mathrm{for}0&lt;x\le 1, \\
{{x}^{2}}-6x+5\mathrm{for}1&lt;x\le 5, \\
\end{matrix} \right.$
</p>
<p>and that $\mathrm{h}\left( x \right)=\mathrm{h}\left( x-5 \right)$ for all real values of $x$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the range of $\mathrm{f}$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Show that ${{\mathrm{g}}^{-1}}$ exists.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find ${{\mathrm{g}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{g}}^{-1}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the exact value of $\mathrm{h}\left( \frac{11}{2} \right)+\mathrm{h}\left( -2 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{h}\left( x \right)$ for $-4\le x\le 6$, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.</div><p style="text-align: right;">[3]</p></li></ol>

$\mathrm{f}\left( x \right)={{x}^{2}}-6x+5$, for $1 < x\le 5$,

$\mathrm{g}\left( x \right)={{x}^{2}}-6x+5$, for $1 < x < 3$,

$\mathrm{h}\left( x \right)=\left\{ \begin{matrix}
\ln x\,\mathrm{for}0&lt;x\le 1, \\
{{x}^{2}}-6x+5\mathrm{for}1&lt;x\le 5, \\
\end{matrix} \right.$


and that $\mathrm{h}\left( x \right)=\mathrm{h}\left( x-5 \right)$ for all real values of $x$.

Sketching Logarithmic Functions and Solving Equations Graphically

Polynomial Functions and Key Features

2022 ACJC Promo Q9

<p>The function $\mathrm{f}$, with domain the set of all real values, is given by</p>
<p style="text-align: center">${\mathrm{f}(x)=\begin{cases} 4-x^2 & \text{for}\hspace{0.5em} 0 &lt; x \leq 2, \\ 2x-4 & \text{for}\hspace{0.5em} 2 &lt; x \leq 4, \end{cases}}$</p>
<p>and that&nbsp;$\mathrm{f}\left( x \right)=\mathrm{f}\left( x+4 \right)$.</p>
<ol class="i-list"><li data-list-item-id="item-1760352687557-610hgbfe5" data-list-item-path="item-1760352687557-610hgbfe5"><div class="list-item-paragraph">Find $\mathrm{f}\left( 45 \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760352688503-t41nancxz" data-list-item-path="item-1760352688503-t41nancxz"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for&nbsp;$-4\le x\le 7$ and state the range of $\mathrm{f}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{\left( x-1 \right)}^{2}}+2$ for&nbsp;$x\in \mathbb{R}$, $0 &lt; x &lt; 1$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760352725886-yoo9mycvf" data-list-item-path="item-1760352725886-yoo9mycvf"><div class="list-item-paragraph">Explain why the composite function $\mathrm{gf}$ does not exist.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760352726641-9dne8y58p" data-list-item-path="item-1760352726641-9dne8y58p"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{fg}\left( x \right)$ and hence, or otherwise, find ${{\left( \mathrm{fg} \right)}^{-1}}\left( \frac{1}{2} \right)$.<span class="inline-marks">[4]</span></div></li></ol>

The function $\mathrm{f}$, with domain the set of all real values, is given by

${\mathrm{f}(x)=\begin{cases} 4-x^2 & \text{for } 0 < x \leq 2, \\ 2x-4 & \text{for } 2 < x \leq 4, \end{cases}}$

and that $\mathrm{f}\left( x \right)=\mathrm{f}\left( x+4 \right)$.

The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{\left( x-1 \right)}^{2}}+2$ for $x\in \mathbb{R}$, $0 < x < 1$.

2023 JPJC P1 Q5

<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

<p>The functions $\mathrm{g}$ and $\mathrm{h}$ are defined as</p>
<p style="text-align: center">$\mathrm{g}:x\to \left\{ \begin{matrix}
4{{\left( x+1 \right)}^{2}}\mathrm{         for }x\in \mathbb{R}\mathrm{,    }-3\le x&lt;0,  \\
4x+4\mathrm{             for }x\in \mathbb{R}\mathrm{,       }0\le x&lt;3.  \\
\end{matrix} \right.$
</p>
<p style="text-align: center">$\mathrm{h}:x\mapsto 2\cos x$,          $x\notin \mathbb{R}$,           $0\le x\le \frac{\pi }{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Prove that the function $\mathrm{gh}$ exists and find the range of $\mathrm{gh}$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Suppose now that $\mathrm{g}\left( x \right)=\mathrm{g}\left( x+6 \right)$ for all real values of $x$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the values of $\mathrm{g}\left( 56 \right)$ and $\mathrm{g}\left( 99 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{g}\left( x \right)$ for $-3\le x\le 8$.</div><p style="text-align: right;">[3]</p></li></ol>

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

$\mathrm{g}:x\to \left\{ \begin{matrix}
4{{\left( x+1 \right)}^{2}}\mathrm{         for }x\in \mathbb{R}\mathrm{,    }-3\le x&lt;0,  \\
4x+4\mathrm{             for }x\in \mathbb{R}\mathrm{,       }0\le x&lt;3.  \\
\end{matrix} \right.$


$\mathrm{h}:x\mapsto 2\cos x$,          $x\notin \mathbb{R}$,           $0\le x\le \frac{\pi }{2}$.

Suppose now that $\mathrm{g}\left( x \right)=\mathrm{g}\left( x+6 \right)$ for all real values of $x$.

2022 CJC J1 MYE Q8

<p>It is given that</p>
<p style="text-align: center">${\mathrm{f}(x) = \begin{cases} 4a^2 - x^2, & \mathrm{for } \ 0 &lt; x \leq 2a, \\ 2a(x - 2a), & \mathrm{for } \ 2a &lt; x \leq 4a, \end{cases}}$</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="i-list"><li data-list-item-id="item-1759608391999-s4e6qpy1c" data-list-item-path="item-1759608391999-s4e6qpy1c"><div class="list-item-paragraph">Evaluate $\mathrm{f}(2019a)$ in terms of $a$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759608392847-nl1e61yf8" data-list-item-path="item-1759608392847-nl1e61yf8"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}(x)$ for&nbsp;$-3a\le x\le 5a$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \sqrt{4{{a}^{2}}-{{(x-2a)}^{2}}}$, ${2a &lt; x &lt; 4a}$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1759608487490-qdi33kkod" data-list-item-path="item-1759608487490-qdi33kkod"><div class="list-item-paragraph">Determine whether the composite function $\mathrm{gf}$ exists, justifying your answer.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759608489834-csenp3uz8" data-list-item-path="item-1759608489834-csenp3uz8"><div class="list-item-paragraph">Give, in terms of $a$, a definition of $\mathrm{fg}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759608490954-az4738wc1" data-list-item-path="item-1759608490954-az4738wc1"><div class="list-item-paragraph">Given that&nbsp;${{\left( \mathrm{fg} \right)}^{-1}}\left( 27 \right)=\frac{7}{2}a$, find the exact value of $a$.<span class="inline-marks">[2]</span></div></li></ol>

${\mathrm{f}(x) = \begin{cases} 4a^2 - x^2, & \mathrm{for } \ 0 < x \leq 2a, \\ 2a(x - 2a), & \mathrm{for } \ 2a < x \leq 4a, \end{cases}}$

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

$\mathrm{g}:x\mapsto \sqrt{4{{a}^{2}}-{{(x-2a)}^{2}}}$, ${2a < x < 4a}$.

2019 JPJC P1 Q7

<ol class="abc-list"><li data-list-item-id="item-1759580130044-o1u80t95o" data-list-item-path="item-1759580130044-o1u80t95o"><div class="list-item-paragraph">It is given that</div><div class="list-item-paragraph" style="text-align: center">${\mathrm{f}(x) = \begin{cases} 2 + \sqrt{9 - (x-3)^2} & \text{for}\hspace{0.5em} 0 &lt; x &lt; 6, \\  \qquad\qquad 2 & \text{for}\hspace{0.5em} 6 \leq x \leq 12. \end{cases}}$</div><div class="list-item-paragraph">and that&nbsp;$\mathrm{f}\left( x \right)=\mathrm{f}\left( x+12 \right)$ for all real values of $x$.</div><ol class="i-list"><li data-list-item-id="subitem-1759580198644-rmjve1hux" data-list-item-path="item-1759580130044-o1u80t95o/subitem-1759580198644-rmjve1hux"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for&nbsp;$-11\le x\le 12$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1759580198790-4lz3vfd62" data-list-item-path="item-1759580130044-o1u80t95o/subitem-1759580198790-4lz3vfd62"><div class="list-item-paragraph">Find the exact value of $\displaystyle\displaystyle \displaystyle\int_{-9}^{10}{\mathrm{f}\left( x \right)\,\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1759580225400-pibm02ob0" data-list-item-path="item-1759580225400-pibm02ob0"><div class="list-item-content"><span class="list-item-text">The curve $C$ has equation $y=\frac{2{{x}^{2}}+3}{{{x}^{2}}-6}$.</span></div><div class="list-item-paragraph">Using an algebraic method, find the set of value of $y$ that&nbsp;$C$ can take.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has equation $y=\frac{2{{x}^{2}}+3}{{{x}^{2}}-6}$.

Math Question Bank