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 $\mathrm{f}$ is defined by \[\mathrm{f}:x\mapsto\sin 2x+\sqrt{3}\cos 2x,\quad 0\leq x\leq k,\] where&nbsp;$k&gt;0$.</p>
<ol class="abc-list"><li data-list-item-id="rf-65" data-list-item-path="rf-65"><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)$, where&nbsp;$R&gt;0$ and $0&lt;\alpha&lt;\frac{\pi}{2}$, giving the exact values of $R$ and $\alpha$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rf-66" data-list-item-path="rf-66"><div class="list-item-paragraph">Find the largest value of $k$ for which $\mathrm{f}^{-1}$ exists. For this value of $k$, find $\mathrm{f}^{-1}(x)$ and state its domain.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="rf-67" data-list-item-path="rf-67"><div class="list-item-paragraph"><p>Using the value of $k$ found in part <strong>(b)</strong>, sketch $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the same diagram. Label the endpoints of both curves and indicate their relationship to the line $y=x$.</p><span class="inline-marks">[3]</span></div></li></ol>
<p>The functions $\mathrm{g}$ and $\mathrm{h}$ are defined by \[\mathrm{g}:x\mapsto\sin 2x+\sqrt{3}\cos 2x,\quad 0&lt;x\leq k,\] \[\mathrm{h}:x\mapsto\frac{1}{x-\sqrt{3}},\quad x&gt;\sqrt{3},\]</p>
<p>where&nbsp;$k$ has the value found in part <strong>(b)</strong>.</p>
<ol class="abc-list" start="4"><li data-list-item-id="rf-70" data-list-item-path="rf-70"><div class="list-item-paragraph">Explain why $\mathrm{hg}$ exists. Find $\mathrm{hg}(x)$ and state its range in exact form.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by \[\mathrm{f}:x\mapsto\sin 2x+\sqrt{3}\cos 2x,\quad 0\leq x\leq k,\] where $k>0$.

Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)$, where $R>0$ and $0<\alpha<\frac{\pi}{2}$, giving the exact values of $R$ and $\alpha$.

Find the largest value of $k$ for which $\mathrm{f}^{-1}$ exists. For this value of $k$, find $\mathrm{f}^{-1}(x)$ and state its domain.

The functions $\mathrm{g}$ and $\mathrm{h}$ are defined by \[\mathrm{g}:x\mapsto\sin 2x+\sqrt{3}\cos 2x,\quad 0<x\leq k,\] \[\mathrm{h}:x\mapsto\frac{1}{x-\sqrt{3}},\quad x>\sqrt{3},\]

where $k$ has the value found in part **(b)**.

Explain why $\mathrm{hg}$ exists. Find $\mathrm{hg}(x)$ and state its range in exact form.

Composite Functions

Sketching a Function and Its Inverse

Existence of Composite Functions

Inverse Trigonometric Functions

Restricting a Domain to Obtain an Inverse

Range of a Composite Function

Inverse Functions

Inverting a shifted sine and forming a reciprocal composite

Functions

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f}: x \mapsto \frac{1}{x^2 + 1}, \quad x \in \mathbb{R}, \: x \le 0.\]</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph">Sketch the graph of $\mathrm{f}$ and explain why the function $\mathrm{f}^{-1}$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Find $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">\[\mathrm{g}: x \mapsto \mathrm{e}^{-x} - 1, \quad x \in \mathbb{R}, \: x \le 1.\]</p>
<ol class="abc-list" start="3"><li data-list-item-id="imported-item-block-5-item-0" data-list-item-path="imported-item-block-5-item-0"><div class="list-item-paragraph">Find the value of $a$ such that&nbsp;$\mathrm{g}^{-1}(a) = 2$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-5-item-1" data-list-item-path="imported-item-block-5-item-1"><div class="list-item-paragraph">Explain why the composite function $\mathrm{gf}$ exists.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-5-item-2" data-list-item-path="imported-item-block-5-item-2"><div class="list-item-paragraph">Find $\mathrm{gf}(x)$ and state the domain of $\mathrm{gf}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1789109821263-d5x6e6ya0" data-list-item-path="item-1789109821263-d5x6e6ya0"><div class="list-item-paragraph"><p>Find the exact range of $\mathrm{gf}$.</p><span class="inline-marks">[1]</span></div></li></ol>

The function $\mathrm{f}$ is defined by

\[\mathrm{f}: x \mapsto \frac{1}{x^2 + 1}, \quad x \in \mathbb{R}, \: x \le 0.\]

Sketch the graph of $\mathrm{f}$ and explain why the function $\mathrm{f}^{-1}$ exists.

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

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

\[\mathrm{g}: x \mapsto \mathrm{e}^{-x} - 1, \quad x \in \mathbb{R}, \: x \le 1.\]

Find the value of $a$ such that $\mathrm{g}^{-1}(a) = 2$.

Explain why the composite function $\mathrm{gf}$ exists.

Find $\mathrm{gf}(x)$ and state the domain of $\mathrm{gf}$.

Horizontal Line Test

2025 YIJC Promo Q11 [Modified]

<ol class="abc-list"><li data-list-item-id="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-0" data-list-item-path="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-0"><div class="list-item-paragraph">The function $\mathrm{f}$ is defined as follows.</div><div class="list-item-paragraph"><p style="text-align: center;">$\mathrm{f}:x\mapsto a(x-1)^2+b,\qquad x\in\mathbb{R},\ x&gt;0.$</p></div><ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-0-sub-0" data-list-item-path="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-0/h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-0-sub-0"><div class="list-item-paragraph">Given that&nbsp;$a&gt;0$ and $0&lt;b&lt;1$, show that&nbsp;$\mathrm{f}^2$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-0-sub-1" data-list-item-path="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-0/h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-0-sub-1"><div class="list-item-paragraph">Hence determine the range of $\mathrm{f}^2$, leaving your answer in terms of $b$.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-1" data-list-item-path="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-1"><div class="list-item-paragraph">The function $\mathrm{g}$ is defined as follows.</div><div class="list-item-paragraph"><p style="text-align: center;">$\mathrm{g}:x\mapsto \frac{x+4}{2x-1},\qquad x\in\mathbb{R},\ x \ne \frac{1}{2}.$</p></div><ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-1-sub-0" data-list-item-path="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-1/h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-1-sub-0"><div class="list-item-paragraph">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="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-1-sub-1" data-list-item-path="h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-1/h2-2025-sajc-promo-q5-sajc-q5-item-block-0-item-1-sub-1"><div class="list-item-paragraph">Find the values of $c$ such that&nbsp;$\mathrm{g}^{2024}(c)=\mathrm{g}^{-1}(c)$. Explain your answers clearly.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Iterates $f^n$ for Large $n$

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

2025 SAJC Promo Q5

<ol class="abc-list"><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-0" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-0"><div class="list-item-paragraph">Given that&nbsp;$a$ is a positive constant, describe a sequence of transformations which transform the graph of $y=e^x$ to the graph of $y=ae^{-x+6}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1"><div class="list-item-paragraph"><p>The function g and h are defined by</p><p>\[\begin{align*} \mathrm{g} &amp;: x \mapsto ae^{-x+6}, \quad x \in \mathbb{R}, \quad a &gt; 0, \\ \mathrm{h} &amp;: x \mapsto 6 + \ln x, \quad x &gt; 0 \end{align*}\]</p></div><ol class="i-list"><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-0" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1/h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-0"><div class="list-item-paragraph"><p>Find the range of $g$.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-1" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1/h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-1"><div class="list-item-paragraph">Hence determine whether the function hg exists.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-2" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1/h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-2"><div class="list-item-paragraph"><p>Given instead that&nbsp;$a=3$.</p></div><ol class="num-list"><li data-list-item-id="item-1789110199479-w9h9bsfum" data-list-item-path="item-1789110199479-w9h9bsfum"><div class="list-item-paragraph"><p>Find $\mathrm{hg}(x)$ and state the domain of $\mathrm{hg}$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1789110210271-d4buk4zf3" data-list-item-path="item-1789110210271-d4buk4zf3"><div class="list-item-paragraph"><p>Find the exact range of $\mathrm{hg}$.</p><span class="inline-marks">[2]</span></div></li></ol></li></ol></li></ol>

Given that $a$ is a positive constant, describe a sequence of transformations which transform the graph of $y=e^x$ to the graph of $y=ae^{-x+6}$.

Composite Transformations

Range of a Function

2025 MI PU1 Promo Q7 [Modified]

Transformations of Graphs

<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 $\mathrm{f}^{-1}(x) = \mathrm{g}\left(\frac{3\pi}{4}\right)$.

Piecewise-Defined Functions

2025 HCI Promo Q8

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{1}{1-x},\ x\in\mathbb{R},\ x\ne 1$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-ejc-promo-q8-ejc-q8-item-block-2-item-0" data-list-item-path="h2-2025-ejc-promo-q8-ejc-q8-item-block-2-item-0"><div class="list-item-paragraph">Explain why $\mathrm{f}^2$ does not exist.<span class="inline-marks">[2]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \frac{1}{1-x},\ x\in\mathbb{R},\ x\ne 0,1$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="h2-2025-ejc-promo-q8-ejc-q8-item-block-5-item-0" data-list-item-path="h2-2025-ejc-promo-q8-ejc-q8-item-block-5-item-0"><div class="list-item-paragraph">Given that&nbsp;$\mathrm{g}^2$ exists, find $\mathrm{g}^2(x)$. State its domain and range.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q8-ejc-q8-item-block-5-item-1" data-list-item-path="h2-2025-ejc-promo-q8-ejc-q8-item-block-5-item-1"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{g}^3(x)=x$ and hence solve the equation $\mathrm{g}^{2025}(x)+\mathrm{g}(x)=0$.<span class="inline-marks">[3]</span></div></li></ol>

$\mathrm{f}:x\mapsto \frac{1}{1-x},\ x\in\mathbb{R},\ x\ne 1$.

Explain why $\mathrm{f}^2$ does not exist.

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

Given that $\mathrm{g}^2$ exists, find $\mathrm{g}^2(x)$. State its domain and range.

Show that $\mathrm{g}^3(x)=x$ and hence solve the equation $\mathrm{g}^{2025}(x)+\mathrm{g}(x)=0$.

2025 EJC Promo Q8

<p>A function $\mathrm{f}$ is said to be self-inverse if&nbsp;$\mathrm{f}(x) = \mathrm{f}^{-1}(x)$ for all $x$ in the domain of $\mathrm{f}$.</p>
<p>It is given that&nbsp;$\mathrm{g}$ is a self-inverse function and is defined by $\mathrm{g} : x \mapsto \frac{x+a}{3x+b}$, for&nbsp;$x \in \mathbb{R},\, x &gt; -\frac{b}{3}$,</p>
<p>where&nbsp;$a$ and $b$ are constants and $\mathrm{g}(1) = 5$.</p>
<ol class="i-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">Show that&nbsp;$a = 9$ and $b = -1$.</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">Find $\mathrm{g}^{2021}(1)$.</div></li></ol>
<p>The function $\mathrm{h}$ is defined by</p>
<p style="text-align: center">\[\mathrm{h} : x \mapsto \left\vert{} 1-x \right\vert{}(x+5),\text{ for}\hspace{0.5em} x \in \mathbb{R},\, x \ge 2.\]</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Given that&nbsp;$\mathrm{h}^{-1}$ exists, find $\mathrm{h}^{-1}$ in similar form.</div></li><li data-list-item-id="imported-item-block-6-item-1" data-list-item-path="imported-item-block-6-item-1"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{gh}$ exists and find the exact range of $\mathrm{gh}$.</div></li></ol>

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

It is given that $\mathrm{g}$ is a self-inverse function and is defined by $\mathrm{g} : x \mapsto \frac{x+a}{3x+b}$, for $x \in \mathbb{R},\, x > -\frac{b}{3}$,

where $a$ and $b$ are constants and $\mathrm{g}(1) = 5$.

The function $\mathrm{h}$ is defined by

\[\mathrm{h} : x \mapsto \left\vert{} 1-x \right\vert{}(x+5),\text{ for } x \in \mathbb{R},\, x \ge 2.\]

Given that $\mathrm{h}^{-1}$ exists, find $\mathrm{h}^{-1}$ in similar form.

Show that $\mathrm{gh}$ exists and find the exact range of $\mathrm{gh}$.

Modulus Functions

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto \frac{ax}{bx - a}, \quad \text{for}\hspace{0.5em} x \in \mathbb{R}, \ x \neq \frac{a}{b},\]</p>
<p>where&nbsp;$a$ and $b$ are non-zero constants.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Find $\mathrm{f}^{-1}(x)$. Hence or otherwise find $\mathrm{f}^2(x)$ and state the range of $\mathrm{f}^2$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">The function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \frac{1}{x}$ for all real non-zero $x$. State whether the composite function $\mathrm{fg}$ exists, justifying your answer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Solve the equation $\mathrm{f}^{-1}(x) = x$.<span class="inline-marks">[3]</span></div></li></ol>

\[\mathrm{f} : x \mapsto \frac{ax}{bx - a}, \quad \text{for } x \in \mathbb{R}, \ x \neq \frac{a}{b},\]

where $a$ and $b$ are non-zero constants.

N2009 P2 Q3

<p>Given that&nbsp;$\mathrm{g}(x) = 2 + \mathrm{e}^x$, $x \in \mathbb{R}$ and $\mathrm{f}(x) = -\ln x$, $x \in \mathbb{R},x&gt;0$.</p>
<p>Find the exact range of</p>
<ol class="abc-list"><li><div class="list-item-paragraph"> $\mathrm{fg}$,</div></li><li><div class="list-item-paragraph"> $\mathrm{gf}$.</div></li></ol>

Given that $\mathrm{g}(x) = 2 + \mathrm{e}^x$, $x \in \mathbb{R}$ and $\mathrm{f}(x) = -\ln x$, $x \in \mathbb{R},x>0$.

<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 $x$.

Graphical Method

Periodic Functions

Inequalities involving Modulus Functions

2024 RI Promo Q10

Inequalities

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p>\[\begin{aligned} &\mathrm{f} : x \mapsto 5 - (x - 2)^2, \quad \text{for}\hspace{0.5em} x \in \mathbb{R}, \ x \ge 0, \\ &\mathrm{g} : x \mapsto \mathrm{e}^x + 3, \qquad \text{for}\hspace{0.5em} x \in \mathbb{R}. \end{aligned}\]</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that the composite function $\mathrm{fg}$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{fg}(x)$ and state its domain.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the range of $\mathrm{fg}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{f}^{-1}$ does not exist.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-v" data-list-item-path="item-v"><div class="list-item-paragraph">If the domain of $\mathrm{f}$ is restricted to $[k, \infty)$, find the smallest value of $k$ such that&nbsp;$\mathrm{f}^{-1}$ exists.<span class="inline-marks">[1]</span></div></li></ol>
<p>For the rest of the question, the domain of $\mathrm{f}$ is now restricted to $[k, \infty)$, with the value of $k$ found in part <strong>(v)</strong>.</p>
<ol class="i-list" start="6"><li data-list-item-id="item-vi" data-list-item-path="item-vi"><div class="list-item-paragraph">Find $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$.<span class="inline-marks">[3]</span></div></li></ol>

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

\[\begin{aligned} &\mathrm{f} : x \mapsto 5 - (x - 2)^2, \quad \text{for } x \in \mathbb{R}, \ x \ge 0, \\ &\mathrm{g} : x \mapsto \mathrm{e}^x + 3, \qquad \text{for } x \in \mathbb{R}. \end{aligned}\]

For the rest of the question, the domain of $\mathrm{f}$ is now restricted to $[k, \infty)$, with the value of $k$ found in part **(v)**.

One-to-One Functions and Inverse Existence

2024 MI PRU1 Q7

<p>A function $\mathrm{h}$ is self-inverse if&nbsp;$\mathrm{h}(x) = \mathrm{h}^{-1}(x)$ for all $x$ in the domain of $\mathrm{h}$.</p>
<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by \[\mathrm{f}(x) = 2 + \frac{1}{x-2}, \quad \text{for}\hspace{0.5em} x \in \mathbb{R}, x &gt; 2\] \[\mathrm{g}(x) = \begin{cases} 3x - 1 & \text{for}\hspace{0.5em} x \le 1, \\ (x - 1)^2 + 2 & \text{for}\hspace{0.5em} x &gt; 1. \end{cases}\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{f}(x)$. With the aid of your graph, explain why $\mathrm{f}$ has an inverse.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{f}$ is self-inverse and find $\mathrm{f}^2(x)$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Hence, or otherwise, evaluate $\mathrm{f}^{2025}(4)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{g}\mathrm{f}$ and state its domain.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Find the range of $\mathrm{g}\mathrm{f}$.<span class="inline-marks">[2]</span></div></li></ol>

A function $\mathrm{h}$ is self-inverse if $\mathrm{h}(x) = \mathrm{h}^{-1}(x)$ for all $x$ in the domain of $\mathrm{h}$.

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by \[\mathrm{f}(x) = 2 + \frac{1}{x-2}, \quad \text{for } x \in \mathbb{R}, x > 2\] \[\mathrm{g}(x) = \begin{cases} 3x - 1 & \text{for } x \le 1, \\ (x - 1)^2 + 2 & \text{for } x > 1. \end{cases}\]

2024 CJC Promo Q11

<p>The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5 + 2x}$, for&nbsp;$x \in \mathbb{R}$, $x &gt; -\frac{5}{2}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the value of $a$ such that&nbsp;$\mathrm{f}^2(a) = 5$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|$, for&nbsp;$x \in \mathbb{R}$, $x \le k$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">State the largest value of $k$ for&nbsp;$\mathrm{g}^{-1}$ to exist.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is now given that&nbsp;$k = -1$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find $\mathrm{g}^{-1}(x)$ and state its domain.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Explain why the composite function $\mathrm{fg}$ exists. State the exact range of $\mathrm{fg}$.<span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5 + 2x}$, for $x \in \mathbb{R}$, $x > -\frac{5}{2}$.

Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|$, for $x \in \mathbb{R}$, $x \le k$.

2024 ASRJC Promo Q7

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="margin-left: 4rem">$\mathrm{f}(x)=1+3\mathrm{e}^{-x}, \quad\quad\quad\, x \in \mathbb{R}, x&gt;0,$ <br>$\mathrm{g}(x)=|x-1|(x-3), \quad x \in \mathbb{R}, x&lt;c \text{ where}\hspace{0.5em} c \text{ is a real constant.}$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$c=4$, determine if the composite functions $\mathrm{fg}$ and $\mathrm{gf}$ exist, justifying your answers. Find the range of the composite function that exists.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$\mathrm{g}^{-1}$ exists, state the largest possible value of $c$. Using this value of $c$ find $\mathrm{g}^{-1}(x)$.</span><span class="inline-marks">[4]</span></div></li></ol>

$\mathrm{f}(x)=1+3\mathrm{e}^{-x}, \quad\quad\quad\, x \in \mathbb{R}, x>0,$ 
$\mathrm{g}(x)=|x-1|(x-3), \quad x \in \mathbb{R}, x<c \text{ where } c \text{ is a real constant.}$

Given that $c=4$, determine if the composite functions $\mathrm{fg}$ and $\mathrm{gf}$ exist, justifying your answers. Find the range of the composite function that exists.

Given that $\mathrm{g}^{-1}$ exists, state the largest possible value of $c$. Using this value of $c$ find $\mathrm{g}^{-1}(x)$.

2020 RI P2 Q3

<p></p>
<ol class="abc-list"><li><div class="list-item-paragraph">A function $\mathrm{g}$ is said to be self-inverse if&nbsp;$\mathrm{g}(x) = \mathrm{g}^{-1}(x)$ for all $x$ in the domain of $\mathrm{g}$.<br>The function $\mathrm{g}$ is defined by\[\mathrm{g} : x \mapsto \frac{\alpha x + \beta}{x - 2}, \quad x \in \mathbb{R}, x \neq 2,\]where&nbsp;$\alpha$ and $\beta$ are constants.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the values of $\alpha$ and $\beta$ for&nbsp;$\mathrm{g}$ to be a self-inverse function.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$\mathrm{g}$ is self-inverse, find the value of $x$ where&nbsp;$\mathrm{g}^{2023}(x) = -\frac{\beta}{2}$.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The functions $\mathrm{f}$ and $\mathrm{h}$ are defined by \[\mathrm{f} : x \mapsto ex + 1, \quad x &gt; 0.\]\[\mathrm{h} : x \mapsto 3x^3 - 9x^2 - 12x + 36, \quad x \geq -2.\]</div><ol class="i-list"><li><div class="list-item-paragraph">Are $\mathrm{f}^2(x)$ and $[\mathrm{f}(x)]^2$ equivalent? Explain your answer.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">By considering $\mathrm{f}^2(x)$ and $\mathrm{f}^3(x)$, show that \[\mathrm{f}^n(x) = \frac{e^{n+1}x + e^n(1 - x) - 1}{e - 1}.\]<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Explain why the composite function $\mathrm{hf}$ exists.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the range of $\mathrm{hf}$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2023 HCI C1 BT Q12

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto \frac{-x - 2}{5x + 1}, \quad x \in \mathbb{R}, x \neq -\frac{1}{5}.\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Explain why both the function $\mathrm{f}^{-1}$ and composite function $\mathrm{f}^2$ exist.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{f}^2(x)$ and state the range of $\mathrm{f}^2$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Determine the solution of the equation $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>The function $g$ is defined by</p>
<p style="text-align: center">\[g : x \mapsto \frac{2x - 2}{5x + 4}, \quad x \in \mathbb{R}, x \neq -\frac{4}{5}, x \neq -\frac{1}{5}, x \neq \frac{2}{5}.\]</p>
<ol class="i-list" start="5"><li><div class="list-item-content"><span class="list-item-text">Verify that&nbsp;$g^2(x) = \mathrm{f}(x)$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$k$ is an integer, find $g^{49}(k)$, giving your answer in terms of $k$.</span><span class="inline-marks">[3]</span></div></li></ol>

\[\mathrm{f} : x \mapsto \frac{-x - 2}{5x + 1}, \quad x \in \mathbb{R}, x \neq -\frac{1}{5}.\]

Explain why both the function $\mathrm{f}^{-1}$ and composite function $\mathrm{f}^2$ exist.

Find $\mathrm{f}^2(x)$ and state the range of $\mathrm{f}^2$.

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

\[g : x \mapsto \frac{2x - 2}{5x + 4}, \quad x \in \mathbb{R}, x \neq -\frac{4}{5}, x \neq -\frac{1}{5}, x \neq \frac{2}{5}.\]

Given that $k$ is an integer, find $g^{49}(k)$, giving your answer in terms of $k$.

Intersections of a Function and Its Inverse

2014 VJC Promo Q8

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto 3\sin \left( 2x - \frac{1}{6}\pi \right), \, 0 \le x \le k\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that the largest exact value of $k$ such that&nbsp;$\mathrm{f}^{-1}$ exists is $\frac{1}{3}\pi$. Find $\mathrm{f}^{-1}(x)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">It is given that&nbsp;$k = \frac{1}{3}\pi$. In a single diagram, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}^{-1}(x)$, labelling your graphs clearly.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{h}$ is defined by $\mathrm{h} : x \mapsto 3\sin \left( 2x - \frac{1}{6}\pi \right), \, 0 \le x \le \frac{1}{3}\pi, \, x \neq \frac{1}{12}\pi$.</p>
<p>Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left| 3 - \frac{1}{x} \right|, \, -3 \le x \le 3, \, x \neq 0$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Explain clearly why $\mathrm{gh}$ exists. Find $\mathrm{gh}(x)$ and its range.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Supposing $(\mathrm{gh})^{-1}$ exists for a restricted domain. Find the exact value of $x$ for which $(\mathrm{gh})^{-1}(x) = 0$. Show your working clearly.<span class="inline-marks">[2]</span></div></li></ol>

\[\mathrm{f} : x \mapsto 3\sin \left( 2x - \frac{1}{6}\pi \right), \, 0 \le x \le k\]

Show that the largest exact value of $k$ such that $\mathrm{f}^{-1}$ exists is $\frac{1}{3}\pi$. Find $\mathrm{f}^{-1}(x)$.

It is given that $k = \frac{1}{3}\pi$. In a single diagram, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}^{-1}(x)$, labelling your graphs clearly.

The function $\mathrm{h}$ is defined by $\mathrm{h} : x \mapsto 3\sin \left( 2x - \frac{1}{6}\pi \right), \, 0 \le x \le \frac{1}{3}\pi, \, x \neq \frac{1}{12}\pi$.

Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left| 3 - \frac{1}{x} \right|, \, -3 \le x \le 3, \, x \neq 0$.

Explain clearly why $\mathrm{gh}$ exists. Find $\mathrm{gh}(x)$ and its range.

Supposing $(\mathrm{gh})^{-1}$ exists for a restricted domain. Find the exact value of $x$ for which $(\mathrm{gh})^{-1}(x) = 0$. Show your working clearly.

2024 DHS Promo Q10

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto \frac{\ln(3x - 2)}{x}, \, x &gt; \frac{2}{3}\]</p>
<p style="text-align: center">\[\mathrm{g} : x \mapsto 3x^2 - 12x + 13, \, x \in \mathbb{R}\]</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = \mathrm{f}(x)$, indicating the equations of asymptotes; coordinates of intersections with the axes and turning points, if any.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{fg}$ exists and find its range.<span class="inline-marks">[2]</span></div></li></ol>
<p>The function $\mathrm{h}$ is defined as</p>
<p style="text-align: center">\[\mathrm{h} : x \mapsto \mathrm{g}(x), \, x \le k\]</p>
<p>such that&nbsp;$\mathrm{h}^{-1}$ exists.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-content"><span class="list-item-text">State the largest value of $k$. For this value of $k$, find $\mathrm{h}^{-1}$ in a similar form.</span><span class="inline-marks">[4]</span></div></li></ol>

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

\[\mathrm{f} : x \mapsto \frac{\ln(3x - 2)}{x}, \, x > \frac{2}{3}\]

\[\mathrm{g} : x \mapsto 3x^2 - 12x + 13, \, x \in \mathbb{R}\]

Sketch the graph of $y = \mathrm{f}(x)$, indicating the equations of asymptotes; coordinates of intersections with the axes and turning points, if any.

The function $\mathrm{h}$ is defined as

\[\mathrm{h} : x \mapsto \mathrm{g}(x), \, x \le k\]

State the largest value of $k$. For this value of $k$, find $\mathrm{h}^{-1}$ in a similar form.

Sketching Logarithmic Functions and Solving Equations Graphically

Asymptotes and Discontinuities

2024 ACJC Promo Q5

Graphing Techniques

<p style="text-align: center">$\mathrm{f}:x\mapsto \cos x,x\in \mathbb{R},0\le x&lt;\frac{\pi }{2}$ and $\mathrm{g}:x\mapsto \ln \left( {{x}^{2}} \right),x\in \mathbb{R},x\ne 0$.</p>
<p>Determine, with reasons, whether $\text{fg}$ and $\text{gf}$ exist. If the composite function exists, give the rule, domain and range in exact form.</p>

$\mathrm{f}:x\mapsto \cos x,x\in \mathbb{R},0\le x<\frac{\pi }{2}$ and $\mathrm{g}:x\mapsto \ln \left( {{x}^{2}} \right),x\in \mathbb{R},x\ne 0$.

Determine, with reasons, whether $\text{fg}$ and $\text{gf}$ exist. If the composite function exists, give the rule, domain and range in exact form.

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by </p>
<p style="text-align: center">$\begin{align}
  & \mathrm{f}:x\mapsto {{x}^{2}}+2x-3,\quad x&gt;-3 \\ 
 & \mathrm{g}:x\mapsto 3x-2,\quad x\ge 0 \\ 
\end{align}$<br></p>
<p>Show that&nbsp;$\mathrm{fg}$ exists and find $\mathrm{fg}$, stating its domain. Find the range of $\mathrm{fg}$. </p>

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

$\begin{align}
  & \mathrm{f}:x\mapsto {{x}^{2}}+2x-3,\quad x>-3 \\ 
 & \mathrm{g}:x\mapsto 3x-2,\quad x\ge 0 \\ 
\end{align}$


Show that $\mathrm{fg}$ exists and find $\mathrm{fg}$, stating its domain. Find the range of $\mathrm{fg}$. 

Math Question Bank