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 $\mathrm{f} : x \mapsto 1 - (x + 3)^2$, $x \in \mathbb{R}$, $x \ge -2$.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Show that the inverse function of $\mathrm{f}$ exists. Find $\mathrm{f}^{-1}(x)$ and state its domain.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">On the same diagram, sketch the graphs of $y = \mathrm{f}(x)$, $y = \mathrm{f}^{-1}(x)$ and $y = \mathrm{f}^{-1}\mathrm{f}(x)$, showing clearly the relationship between the graphs.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Hence, find the exact solution of $\mathrm{f}^{-1}\mathrm{f}(x) \le \mathrm{f}(x)$.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto 1 - (x + 3)^2$, $x \in \mathbb{R}$, $x \ge -2$.

Composite Functions

Sketching a Function and Its Inverse

One-to-One Functions and Inverse Existence

Sketching $y=f(f^{-1}(x))$ and $y=f^{-1}(f(x))$

Inverse Functions

2011 NJC P1 Q7

Functions

<p>The function $\mathrm{f}$ is defined by $\mathrm{f}:x \mapsto \frac{1}{2}+e^{-x}$, $x \in \mathbb{R}$, $x \ge 0$.</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 $y=\mathrm{f}(x)$, $y=\mathrm{f}^{-1}(x)$ and $y=\mathrm{f}^{-1}\mathrm{f}(x)$ on the same diagram, showing clearly their relationship.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by $\mathrm{g}:x \mapsto 1+4x-x^2$, $x \le 2$.</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">Explain why $\mathrm{g}^{-1}$ exists.<span class="inline-marks">[1]</span></div></li><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 $\mathrm{gf}$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Find the exact range of $\mathrm{gf}$.<span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f}:x \mapsto \frac{1}{2}+e^{-x}$, $x \in \mathbb{R}$, $x \ge 0$.

Sketch the graph of $y=\mathrm{f}(x)$, $y=\mathrm{f}^{-1}(x)$ and $y=\mathrm{f}^{-1}\mathrm{f}(x)$ on the same diagram, showing clearly their relationship.

The function $\mathrm{g}$ is defined by $\mathrm{g}:x \mapsto 1+4x-x^2$, $x \le 2$.

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

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

Existence of Composite Functions

Range of a Composite Function

2025 TJC P2 Q3

<p>The function $\mathrm{f}$ is defined by</p>
<p>\[\mathrm{f} : x \mapsto \mathrm{e}^x + \frac{1}{2x + 2} \text{ for}\hspace{0.5em} x \in \mathbb{R}, x \neq -1.\]</p>
<p>A function $\mathrm{g}$, defined for&nbsp;$x \in \mathbb{R}, x \geq 1$, is such that&nbsp;$y \to \infty$ as&nbsp;$x$ increases. It is also given that&nbsp;$\mathrm{g}(1) = -0.5$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Explain why the composite function $\mathrm{fg}$ exists and find the corresponding range of $\mathrm{fg}$.<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">Given that&nbsp;$\mathrm{fg}(x) = \frac{x}{\sqrt{\mathrm{e}}} + \frac{1}{2 \ln x + 1}$, find an expression for&nbsp;$\mathrm{g}(x)$ in terms of $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The domain of $\mathrm{f}$ is now further restricted to $x &gt; k$. State the least value of integer $k$ for which the function $\mathrm{f}^{-1}$ exists.<span class="inline-marks">[1]</span></div></li></ol>
<p>For the rest of the question, use the value of $k$ found in part <strong>(c)</strong>.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Without finding $\mathrm{f}^{-1}$,</div><ol class="i-list"><li data-list-item-id="item-d-i" data-list-item-path="item-d/item-d-i"><div class="list-item-paragraph">sketch, on the same diagram, the graphs of $\mathrm{f}$, $\mathrm{f}^{-1}$ and $\mathrm{f}\mathrm{f}^{-1}$ showing clearly the relationships between the graphs,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d-ii" data-list-item-path="item-d/item-d-ii"><div class="list-item-paragraph">find the gradient of the tangent to the graph of $y = \mathrm{f}^{-1}(x)$ at $x = \mathrm{e} + \frac{1}{4}$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

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

\[\mathrm{f} : x \mapsto \mathrm{e}^x + \frac{1}{2x + 2} \text{ for } x \in \mathbb{R}, x \neq -1.\]

A function $\mathrm{g}$, defined for $x \in \mathbb{R}, x \geq 1$, is such that $y \to \infty$ as $x$ increases. It is also given that $\mathrm{g}(1) = -0.5$.

Explain why the composite function $\mathrm{fg}$ exists and find the corresponding range of $\mathrm{fg}$.

Given that $\mathrm{fg}(x) = \frac{x}{\sqrt{\mathrm{e}}} + \frac{1}{2 \ln x + 1}$, find an expression for $\mathrm{g}(x)$ in terms of $x$.

The domain of $\mathrm{f}$ is now further restricted to $x > k$. State the least value of integer $k$ for which the function $\mathrm{f}^{-1}$ exists.

For the rest of the question, use the value of $k$ found in part **(c)**.

Derivative as the Gradient of a Curve

Restricting a Domain to Obtain an Inverse

Recovering a Function from a Composite

2025 DHS P1 Q7

Applications of Differentiation

<p>Given that&nbsp;${{D}_{{{\mathrm{f}}^{-1}}}}=[3,\infty )$and ${{R}_{{{\mathrm{f}}^{-1}}}}=(-5,7)$.	<br>Sketch $\mathrm{f}{{\mathrm{f}}^{-1}}(x)$and${{\mathrm{f}}^{-1}}\mathrm{f}(x)$.<br></p>

Given that ${{D}_{{{\mathrm{f}}^{-1}}}}=[3,\infty )$and ${{R}_{{{\mathrm{f}}^{-1}}}}=(-5,7)$.	
Sketch $\mathrm{f}{{\mathrm{f}}^{-1}}(x)$and${{\mathrm{f}}^{-1}}\mathrm{f}(x)$.


<p>Given that&nbsp;${{D}_{\mathrm{f}}}=(-\infty ,\ln 2)$and ${{R}_{\mathrm{f}}}=(0,5)$.	<br>Sketch $\mathrm{f}{{\mathrm{f}}^{-1}}(x)$ and ${{\mathrm{f}}^{-1}}\mathrm{f}(x)$.</p>

Given that ${{D}_{\mathrm{f}}}=(-\infty ,\ln 2)$and ${{R}_{\mathrm{f}}}=(0,5)$.	
Sketch $\mathrm{f}{{\mathrm{f}}^{-1}}(x)$ and ${{\mathrm{f}}^{-1}}\mathrm{f}(x)$.

<p>Given that&nbsp;$ \mathrm{D}_{\mathrm{f}} = [0, 4) $ and $\mathrm{R}_{\mathrm{f}} = (1, 9]$.</p>
<p>Sketch $\mathrm{f f}^{-1}(x)$ and $\mathrm{f}^{-1} \mathrm{f}(x)$.</p>

Given that $ \mathrm{D}_{\mathrm{f}} = [0, 4) $ and $\mathrm{R}_{\mathrm{f}} = (1, 9]$.

Sketch $\mathrm{f f}^{-1}(x)$ and $\mathrm{f}^{-1} \mathrm{f}(x)$.

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \ln \left( {{x}^{2}}-4 \right)$ for&nbsp;$2 &lt; x\le 8$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch the graphs of $\mathrm{f}$, ${{\mathrm{f}}^{-1}}$ and $\mathrm{f}\,{{\mathrm{f}}^{-1}}$ on the same axes, showing clearly the relationship between the graphs.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The region bounded by the curve $y=\mathrm{f}\left( x \right)$, the line $x=8$ and the $x$-axis is rotated through $2\pi $ radians about the $y$- axis. Find the exact volume of the solid obtained.<span class="inline-marks">[4]</span></div></li></ol>

$\mathrm{f}:x\mapsto \ln \left( {{x}^{2}}-4 \right)$ for $2 < x\le 8$.

2022 TJC Promo Q9

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

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

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

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

Intersections of a Function and Its Inverse

Modulus Functions

2023 ASRJC P2 Q4

<p>The function $\mathrm{f}$ is given by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{4}{1+{{x}^{2}}}$, $x\in \mathbb{R}$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that ${{\mathrm{f}}^{-1}}$ does not exist.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that the largest possible domain for which $\mathrm{f}$ is a one-one function is the set $\left\{ x:x\in \mathbb{R},x\le A \right\}$, state the value of $A$.</div><p style="text-align: right;">[1]</p></li></ol>
<p>In the rest of the question, the domain of $\mathrm{f}$ is $x\in \mathbb{R}$, $x\le A$, where $A$ is the value found in part <strong>(ii)</strong>.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and write down the domain of ${{\mathrm{f}}^{-1}}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">On the same diagram, sketch the graphs of $y=\mathrm{f}\left( x \right)$, $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y=\mathrm{f}{{\mathrm{f}}^{-1}}\left( x \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The composite function $\mathrm{f}\,\mathrm{h}$ is defined by $\mathrm{f}\,\mathrm{h}:x\mapsto {{x}^{2}}$, $0 < x\le 2$. Find $\mathrm{h}\left( x \right)$ and state its domain.</div><p style="text-align: right;">[2]</p></li></ol>

$\mathrm{f}:x\mapsto \frac{4}{1+{{x}^{2}}}$, $x\in \mathbb{R}$

In the rest of the question, the domain of $\mathrm{f}$ is $x\in \mathbb{R}$, $x\le A$, where $A$ is the value found in part **(ii)**.

2024 JPJC Functions Assignment 4 Q1

<p>Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{1}{x-2},$ $x\in \mathbb{R},$ $x\ne 2,$</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto {{x}^{2}}-4x$, $x\in \mathbb{R}.$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Determine whether the composite functions ${{\mathrm{f}}^{2}}$, $\mathrm{fg}$ and $\mathrm{gf}$ exist.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Give a definition (including the domain) of the composite function that exists, and find the range of this composite function.</div><p style="text-align: right;">[2]</p></li></ol>
<p>In the rest of the equation, the domain of $\mathrm{f}$ is $x\in \mathbb{R},$ $x < 2$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Define ${{\mathrm{f}}^{-1}}$ in similar form.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Sketch on a single clearly labelled diagram, the graphs of $y=\mathrm{f}\left( x \right)$, $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)$. Your sketch should indicate the position of the graphs in relation to the origin.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Without using a graphing calculator, solve the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.</div><p style="text-align: right;">[2]</p></li></ol>

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

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

$\mathrm{g}:x\mapsto {{x}^{2}}-4x$, $x\in \mathbb{R}.$

In the rest of the equation, the domain of $\mathrm{f}$ is $x\in \mathbb{R},$ $x < 2$.

2018 MI Promo Q10

<ol class="abc-list"><li><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">$\mathrm{f}:x\mapsto \frac{1}{1+x}-2$, for $x\in \mathbb{R}$, $x\ge 0$,</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{g}:x\mapsto {{\left( x+\frac{4}{3} \right)}^{2}}-\frac{1}{5}$ for $x\in \mathbb{R}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Determine whether the composite function $\mathrm{fg}$ exists.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ and find the range of $\mathrm{f}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the exact range of $\mathrm{gf}$.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">The function $\mathrm{h}$ is defined by</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{h}:x\mapsto {{\left( x+1 \right)}^{2}}\left( x-2 \right)$, $x\in \mathbb{R}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Explain why the function ${{\mathrm{h}}^{-1}}$ does not exist.</div><p style="text-align: right;">[1]</p><div class="list-item-paragraph">For the rest of the question, the domain of $\mathrm{h}$ is restricted to $x\in \left[ -1,1 \right]$.</div></li><li><div class="list-item-paragraph">State the domain of ${{\mathrm{h}}^{-1}}$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Sketch on the same diagram the graphs of $y=\mathrm{h}\left( x \right)$, $y={{\mathrm{h}}^{-1}}\left( x \right)$ and $y={{\mathrm{h}}^{-1}}\mathrm{h}\left( x \right)$.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Range of a Function

2020 NJC Promo Q9

<ol class="abc-list"><li><div class="list-item-paragraph">The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto {{x}^{2}}+4x-4$ for&nbsp;$x\in \mathbb{R}$, $x&lt;-2$.</div><ol class="i-list"><li><div class="list-item-paragraph">Explain why ${{\mathrm{f}}^{-1}}$ exists.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Sketch on the same diagram, the graphs of $y=\mathrm{f}\left( x \right)$, $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\,\mathrm{f}\left( x \right)$, stating the coordinates of endpoints and equations of asymptotes, if any.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Describe the relationship between the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$.<span class="inline-marks">[1]</span></div></li></ol></li><li><div class="list-item-paragraph">The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \ln \left( x+10 \right)$, where&nbsp;$x&gt;-10$.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that the composite function $\mathrm{gf}$ exists. Find the range of $\mathrm{gf}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Solve $\mathrm{f}\left( x \right)={{\mathrm{g}}^{-1}}\left( \ln \left( 6 \right) \right)$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Polynomial Functions and Key Features

2022 SAJC Promo Q6

Graphing Techniques

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \sqrt{3}\cos x-\sin x$, $x\in \mathbb{R}$, $-\pi \le x\le \pi $.</p>
<ol class="i-list"><li data-list-item-id="item-1759741068643-w4x2kdtco" data-list-item-path="item-1759741068643-w4x2kdtco"><div class="list-item-paragraph">Express $\mathrm{f}\left( x \right)$ as&nbsp;$R\cos \left( x+\alpha \right)$, where&nbsp;$R$ and $\alpha $ are constants to be found.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759741069566-omrmg11f2" data-list-item-path="item-1759741069566-omrmg11f2"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, stating clearly the exact coordinates of the end-points and turning points. Hence explain why ${{\mathrm{f}}^{-1}}\left( x \right)$ does not exist.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759741070556-l5qv1eyvg" data-list-item-path="item-1759741070556-l5qv1eyvg"><div class="list-item-paragraph">If the domain of $\mathrm{f}$ is further restricted to $-\frac{\pi }{6}\le x\le k$, state the maximum value of $k$ for which ${{\mathrm{f}}^{-1}}$ exist.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759741071831-qnlad5wfu" data-list-item-path="item-1759741071831-qnlad5wfu"><div class="list-item-paragraph">Using the value of $k$ found in part <strong>(iii)</strong>, find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Hence sketch, on the same diagram, the graphs of $y=\mathrm{f}\left( x \right)$, $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}\left( x \right)=\frac{1}{1-x}$, $x\in \mathbb{R}$, $x\ne 0$, $x\ne 1$.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1759741238340-tvxkyqgol" data-list-item-path="item-1759741238340-tvxkyqgol"><div class="list-item-paragraph">Given that&nbsp;${{\mathrm{g}}^{2}}\left( x \right)=\frac{x-1}{x}$ and ${{\mathrm{g}}^{3}}\left( x \right)=x$ for&nbsp;$x\in \mathbb{R}$, $x\ne 0$, $x\ne 1$, determine the value of ${{\mathrm{g}}^{2018}}\left( 2 \right)$.<span class="inline-marks">[2]</span></div></li></ol>

$\mathrm{f}:x\mapsto \sqrt{3}\cos x-\sin x$, $x\in \mathbb{R}$, $-\pi \le x\le \pi $.

$\mathrm{g}\left( x \right)=\frac{1}{1-x}$, $x\in \mathbb{R}$, $x\ne 0$, $x\ne 1$.

Inverse Trigonometric Functions

Iterates $f^n$ for Large $n$

Horizontal Line Test

2018 MI P1 Q8

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto 3+2x-{{x}^{2}}$, $x\in \mathbb{R}$, $x&gt;1$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">On the same axes, sketch the graphs of $y=\mathrm{f}\left( x \right)$, $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the exact value of $x$ that satisfies the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \sqrt{4-x}$, $x\in \mathbb{R}$, $x\le 4$</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Explain why the composite function $\mathrm{gf}$ exists and find $\mathrm{gf}\left( x \right)$, simplifying your answer.<span class="inline-marks">[3]</span></div></li></ol>

$\mathrm{f}:x\mapsto 3+2x-{{x}^{2}}$, $x\in \mathbb{R}$, $x>1$

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

$\mathrm{g}:x\mapsto \sqrt{4-x}$, $x\in \mathbb{R}$, $x\le 4$

2023 YIJC Promo Q9

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{1}{\left| 1-{{x}^{2}} \right|}$, $x\in \mathbb{R}$, $-2\le x&lt;-1$,</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto -{{\left( x-2 \right)}^{2}}+k$, $x\in \mathbb{R}$, $x\ge 0$ where&nbsp;$k$ is a constant.</p>
<ol class="i-list"><li data-list-item-id="item-1759659554437-8jkl0fw85" data-list-item-path="item-1759659554437-8jkl0fw85"><div class="list-item-paragraph">Sketch on the same diagram the graphs of</div><ol class="abc-list"><li data-list-item-id="subitem-1759659564007-6nk7om3uj" data-list-item-path="item-1759659554437-8jkl0fw85/subitem-1759659564007-6nk7om3uj"><div class="list-item-paragraph">$y=\mathrm{f}\left( x \right)$</div></li><li data-list-item-id="subitem-1759659565312-u4im8280e" data-list-item-path="item-1759659554437-8jkl0fw85/subitem-1759659565312-u4im8280e"><div class="list-item-paragraph">$y={{\mathrm{f}}^{-1}}\left( x \right)$</div></li><li data-list-item-id="subitem-1759659566077-1m7fo5yww" data-list-item-path="item-1759659554437-8jkl0fw85/subitem-1759659566077-1m7fo5yww"><div class="list-item-paragraph">$y={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)$</div></li></ol><div class="list-item-after-content">stating the equations of any asymptotes and the coordinates of any endpoints.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759659591262-23x8jlslp" data-list-item-path="item-1759659591262-23x8jlslp"><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}$ and state the domain of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759659600982-fjm13192h" data-list-item-path="item-1759659600982-fjm13192h"><div class="list-item-paragraph">Show that the composite function $\mathrm{gf}$ exists and find its range.<span class="inline-marks">[2]</span></div></li></ol>

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

$\mathrm{f}:x\mapsto \frac{1}{\left| 1-{{x}^{2}} \right|}$, $x\in \mathbb{R}$, $-2\le x<-1$,

$\mathrm{g}:x\mapsto -{{\left( x-2 \right)}^{2}}+k$, $x\in \mathbb{R}$, $x\ge 0$ where $k$ is a constant.

Sketching Rational Functions

Asymptotes and Discontinuities

Graphs of Modulus Function

2021 ACJC Promo Q7

Transformations of Graphs

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\to {{\left( \frac{3}{x-2} \right)}^{2}}$, $x\in \mathbb{R}$, $x>2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that ${{\mathrm{f}}^{-1}}$ exists and define it in similar form.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Using a new set of axes, sketch the graphs of $\mathrm{f}$, ${{\mathrm{f}}^{-1}}$ and ${{\mathrm{f}}^{-1}}\mathrm{f}$ on the same diagram.</div><p style="text-align: right;">[3]</p></li></ol>

$\mathrm{f}:x\to {{\left( \frac{3}{x-2} \right)}^{2}}$, $x\in \mathbb{R}$, $x>2$.

2022 CJC J1 MYE Q7

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto -\ln (x-2)$, $x\in \mathbb{R}$, $2 &lt; x\le 3$.</p>
<ol class="i-list"><li data-list-item-id="item-1759653487481-isrn9hn87" data-list-item-path="item-1759653487481-isrn9hn87"><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}(x)$ and state the domain and range of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759653488088-g3z7msolq" data-list-item-path="item-1759653488088-g3z7msolq"><div class="list-item-paragraph">Sketch on the same diagram the graphs of $y=\mathrm{f}(x)$, $y={{\mathrm{f}}^{-1}}(x)$ and $y={{\mathrm{f}}^{-1}}\mathrm{f}(x)$, giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the $x$- and $y$- axes.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-1759653488643-tbnrpwn7y" data-list-item-path="item-1759653488643-tbnrpwn7y"><div class="list-item-paragraph">Explain why the x-coordinate of the point of intersection of the graph of $y=\mathrm{f}(x)$ and $y={{\mathrm{f}}^{-1}}(x)$ satisfies the equation</div><div class="list-item-paragraph" style="text-align: center">$x+\ln (x-2)=0$,</div><div class="list-item-paragraph">and find the value of this $x$-coordinate.<span class="inline-marks">[2]</span></div></li></ol>

$\mathrm{f}:x\mapsto -\ln (x-2)$, $x\in \mathbb{R}$, $2 < x\le 3$.

Sketching Exponential Functions and Solving Equations Graphically

Sketching Logarithmic Functions and Solving Equations Graphically

2020 TMJC Promo Q7

<p>The function $\mathrm{h}$ is defined by $\mathrm{h}:x\mapsto \left| \ln \left( 2x \right) \right|+1$, $x\in \mathbb{R}$, $0 &lt; x\le \lambda $.</p>
<ol class="i-list"><li data-list-item-id="item-1759585612740-pz80peu9m" data-list-item-path="item-1759585612740-pz80peu9m"><div class="list-item-paragraph">Find the maximum value of $\lambda $ for which the inverse function $\mathrm{h}$ exist.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759585624792-wps6l2s7o" data-list-item-path="item-1759585624792-wps6l2s7o"><div class="list-item-paragraph">Using $\lambda =\frac{1}{2}$,</div><ol class="abc-list"><li data-list-item-id="subitem-1759585627463-trtwmm7tz" data-list-item-path="item-1759585624792-wps6l2s7o/subitem-1759585627463-trtwmm7tz"><div class="list-item-paragraph">Find ${{\mathrm{h}}^{-1}}$ and state its domain,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1759585627619-0kh5kw7ft" data-list-item-path="item-1759585624792-wps6l2s7o/subitem-1759585627619-0kh5kw7ft"><div class="list-item-paragraph">sketch the graph of $y=\mathrm{h}{{\mathrm{h}}^{-1}}\left( x \right)$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1759585627770-bu5ufpnqq" data-list-item-path="item-1759585624792-wps6l2s7o/subitem-1759585627770-bu5ufpnqq"><div class="list-item-paragraph">find the solution set for&nbsp;$\mathrm{h}\left( x \right)={{\mathrm{h}}^{-1}}\left( x \right)$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

The function $\mathrm{h}$ is defined by $\mathrm{h}:x\mapsto \left| \ln \left( 2x \right) \right|+1$, $x\in \mathbb{R}$, $0 < x\le \lambda $.

Find the maximum value of $\lambda $ for which the inverse function $\mathrm{h}$ exist.

2022 ACJC P1 Q6

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f : }x\mapsto {{\mathrm{e}}^{x-2}}-3,\ \ \ \mathrm{ }x\in \mathbb{R}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find ${{\mathrm{f}}^{-1}}(x)$ and write down the domain and range of ${{\mathrm{f}}^{-1}}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Sketch on the same diagram the graphs of $y=\mathrm{f}(x)$, $y={{\mathrm{f}}^{-1}}(x)$ and $y=\mathrm{f}{{\mathrm{f}}^{-1}}(x)$, giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the $x$- and $y$-axes.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the area of the region bounded between $y=\mathrm{f}(x)$ and $y={{\mathrm{f}}^{-1}}(x)$, giving your answer correct to $2$ decimal places.</span></div><p style="text-align: right;">[2]</p></li></ol>

$\mathrm{f : }x\mapsto {{\mathrm{e}}^{x-2}}-3,\ \ \ \mathrm{ }x\in \mathbb{R}$.

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

Sketch on the same diagram the graphs of $y=\mathrm{f}(x)$, $y={{\mathrm{f}}^{-1}}(x)$ and $y=\mathrm{f}{{\mathrm{f}}^{-1}}(x)$, giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the $x$- and $y$-axes.

Find the area of the region bounded between $y=\mathrm{f}(x)$ and $y={{\mathrm{f}}^{-1}}(x)$, giving your answer correct to $2$ decimal places.

Area Bounded by Curves

2015 IJC P1 Q6

Applications of Integration

<p>Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="text-align: center">$\begin{matrix} & \mathrm{f}:x\mapsto -\ln \left( x+a \right),x\in \mathbb{R},x>-a \\ & \mathrm{g}:x\mapsto {{\left( x-\sqrt{a} \right)}^{2}},x\in \mathbb{R},x>0 \\ \end{matrix}$</p>
<p>where $a$ is a positive constant such that $1 < a\le 3$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y=\mathrm{f}\left( x \right)$ and show that $\mathrm{f}$ has an inverse.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">On the same diagram in part (i), sketch the graph of $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the range of the composite function $\mathrm{fg}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Without finding ${{\mathrm{f}}^{-1}}\left( x \right)$, find $x$ given that ${{\mathrm{f}}^{-1}}\mathrm{g}\left( x \right)={{\mathrm{e}}^{-3}}-a$.</span></div><p style="text-align: right;">[3]</p></li></ol>

$\begin{matrix} & \mathrm{f}:x\mapsto -\ln \left( x+a \right),x\in \mathbb{R},x>-a \\ & \mathrm{g}:x\mapsto {{\left( x-\sqrt{a} \right)}^{2}},x\in \mathbb{R},x>0 \\ \end{matrix}$

where $a$ is a positive constant such that $1 < a\le 3$.

Sketch the graph of $y=\mathrm{f}\left( x \right)$ and show that $\mathrm{f}$ has an inverse.

On the same diagram in part (i), sketch the graph of $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)$.

Find the range of the composite function $\mathrm{fg}$.

Without finding ${{\mathrm{f}}^{-1}}\left( x \right)$, find $x$ given that ${{\mathrm{f}}^{-1}}\mathrm{g}\left( x \right)={{\mathrm{e}}^{-3}}-a$.

Math Question Bank