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

<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>

Existence of Composite Functions

Range of a Composite Function

Iterates $f^n$ for Large $n$

Inverse Functions

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

2025 SAJC Promo Q5

Functions

<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

Inverse Trigonometric Functions

Restricting a Domain to Obtain an Inverse

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>

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

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

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

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

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

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 $\mathrm{g}^{2025}(x)+\mathrm{g}(x)=0$.

Composite Functions

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

Horizontal Line Test

2024 CJC Promo Q11

<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

One-to-One Functions and Inverse Existence

2014 VJC Promo Q8

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{1}{1-x}$,	$x\in \mathbb{R}$, $x\ne 1$, $x\ne 0$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;${{\mathrm{f}}^{2}}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.		<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{3}}\left( x \right)$ and ${{\mathrm{f}}^{2020}}\left( x \right)$ in simplest form.		<span class="inline-marks">[3]</span></div></li></ol>

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

N2014 P1 Q1 [Modified]

<p>The function $\mathrm{f}$ is defined by</p>
<p>\[\mathrm{f} : x \mapsto \frac{x-1}{x}, x \in \mathbb{R}, x \neq 0, x \neq 1\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$\mathrm{f}^2(x) = \mathrm{f}^{-1}(x)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{f}^3(x)$ in simplified form.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{f}^{2030}(5)$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Functions $\mathrm{g}$ and $\mathrm{h}$ are defined by</p>
<p>\[\mathrm{g} : x \mapsto \frac{x-1}{x}, x \in \mathbb{R}, x \geq 1\]</p>
<p>\[\mathrm{h} : x \mapsto -\sin ax, x \in \mathbb{R}\]</p>
<p>where&nbsp;$a$ is a positive constant.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the value of $a$ given that the range of $\mathrm{hg}$ is $(-1, 0]$.</span><span class="inline-marks">[2]</span></div></li></ol>

\[\mathrm{f} : x \mapsto \frac{x-1}{x}, x \in \mathbb{R}, x \neq 0, x \neq 1\]

Show that $\mathrm{f}^2(x) = \mathrm{f}^{-1}(x)$.

Find $\mathrm{f}^3(x)$ in simplified form.

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

\[\mathrm{g} : x \mapsto \frac{x-1}{x}, x \in \mathbb{R}, x \geq 1\]

\[\mathrm{h} : x \mapsto -\sin ax, x \in \mathbb{R}\]

Find the value of $a$ given that the range of $\mathrm{hg}$ is $(-1, 0]$.

2025 VJC P1 Q2

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

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

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

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

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

Range of a Function

2025 HCI P1 Q8

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:</p>
<p style="text-align: center">$\mathrm{f}: x \mapsto \dfrac{5 - x}{1 - x}, x \in \mathbb{R}, x \ne 1$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain why $\mathrm{f}$ has an inverse, $\mathrm{f}^{-1}$, and show that&nbsp;$\mathrm{f}^{-1} = \mathrm{f}$.<br><span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Evaluate $\mathrm{f}^{51}(4)$.<span class="inline-marks">[2]</span></div></li></ol>

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

$\mathrm{f}: x \mapsto \dfrac{5 - x}{1 - x}, x \in \mathbb{R}, x \ne 1$

2009 HCI P1 Q9

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto -\frac{x}{2x+1},x\in \mathbb{R},x\ne -\frac{1}{2}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760955728376-m71ouhwhw" data-list-item-path="item-1760955728376-m71ouhwhw"><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="item-1760955728889-plghdz398" data-list-item-path="item-1760955728889-plghdz398"><div class="list-item-paragraph">Hence, or otherwise, find ${{\mathrm{f}}^{2}}\left( x \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760955758360-czlenietc" data-list-item-path="item-1760955758360-czlenietc"><div class="list-item-paragraph">Deduce the exact value of ${{\mathrm{f}}^{2023}}\left( 5 \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \frac{1}{ax},x\in \mathbb{R},x&gt;a$,</p>
<p>where&nbsp;$a$ is a positive constant.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760955790730-bfc1j31hg" data-list-item-path="item-1760955790730-bfc1j31hg"><div class="list-item-paragraph">Determine whether the composite function $\mathrm{fg}$ exists, justifying your answer.<span class="inline-marks">[2]</span></div></li></ol>

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

$\mathrm{g}:x\mapsto \frac{1}{ax},x\in \mathbb{R},x>a$,

2023 MI P1 Q6

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

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

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

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

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

2024 HCI Promo Q8

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{3-2x}{x+2}$, for $x\in \mathbb{R}$, $x>-2$,</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \left| x \right|-1$, for $x\in \mathbb{R}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state its domain.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find ${{\mathrm{f}}^{2024}}\left( 1 \right)$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Show that $\mathrm{fg}$ exists and find its range.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Solve $\mathrm{f}\left( x \right)>\mathrm{g}\left( x \right)$, giving your answer in exact form.</div><p style="text-align: right;">[4]</p></li></ol>

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

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

$\mathrm{g}:x\mapsto \left| x \right|-1$, for $x\in \mathbb{R}$.

2024 CJC P2 Q2

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

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

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

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

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

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

2024 SAJC P2 Q3

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

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

2024 DHS P2 Q4

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

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

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

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

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

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

Sketching Exponential Functions and Solving Equations Graphically

Asymptotes and Discontinuities

2013 HCI C2 BT2 P1 Q11 [Modified]

Graphing Techniques

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{2x+k}{x-2},x\in \mathbb{R},\mathrm{ }x\ne 2$</p>
<p>where&nbsp;$k$ is a positive constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, stating the equations of any asymptotes and the coordinates of any points where the curve crosses the $x$ and $y$ axes.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Describe fully a sequence of transformations which would transform the curve $y=\frac{1}{x}$ onto $y=\mathrm{f}\left( x \right)$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}$ in a similar form and write down the range of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence or otherwise, find ${{\mathrm{f}}^{2}}$.</div><div class="list-item-paragraph">Find the value of ${{\mathrm{f}}^{2017}}\left( \frac{1}{2} \right)$, leaving your answer in terms of $k$.<span class="inline-marks">[4]</span></div></li></ol>

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

Composite Transformations

Sketching Rational Functions

2016 CJC P1 Q11

Transformations of Graphs

<p>The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \frac{x+2}{x-1}$, for $x\in \mathbb{R}$, $x\ne 1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{2}}(x)$ and ${{\mathrm{f}}^{2012}}(x)$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \cos x$, for $0 < x < 2\pi $.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Explain why the composite function $\mathrm{fg}$ exists.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Define $\mathrm{fg}$, giving its domain.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the range of $\mathrm{fg}$.</div><p style="text-align: right;">[1]</p></li></ol>

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

The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \cos x$, for $0 < x < 2\pi $.

2012 JJC Promo Q5

<p>The function $\mathrm{g}$, with domain $\{x\in \mathbb{R}:1\le x\le 6\}$, has six of its function values given in the table below.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760094776400-gm39f3gi44u.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAwOTQ3NzY0MDAtZ20zOWYzZ2k0NHUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDc5MSwiZXhwIjoxNzkzNTA1MDkxfQ.oVKB7taIbDc4-0k1fRNiccWaeMXGOh-DizhpSl3jh-w" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">Use the table to explain why $\mathrm{g}$ does not have an inverse function.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find ${{\mathrm{g}}^{3}}\left( 3 \right)$. Hence find the set of all positive integers $n$ for which ${{\mathrm{g}}^{n}}\left( 3 \right)=4$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>It is also known that for $k=1$, $2$, $3$, $4$, $5$</p>
<p style="text-align: center">$\mathrm{g}\left( x \right)=\mathrm{g}\left( k \right)+\left( \mathrm{g}\left( k+1 \right)-\mathrm{g}\left( k \right) \right)\left( x-k \right)$ for $k < x < k+1$</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Evaluate $\mathrm{g}\left( 1.5 \right)$and $\mathrm{g}\left( 2.7 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{g}\left( x \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the range of values of $k$ for which the equation $\mathrm{g}\left( x \right)=k$ has four real distinct roots.</div><p style="text-align: right;">[1]</p></li></ol>

The function $\mathrm{g}$, with domain $\{x\in \mathbb{R}:1\le x\le 6\}$, has six of its function values given in the table below.

It is also known that for $k=1$, $2$, $3$, $4$, $5$

$\mathrm{g}\left( x \right)=\mathrm{g}\left( k \right)+\left( \mathrm{g}\left( k+1 \right)-\mathrm{g}\left( k \right) \right)\left( x-k \right)$ for $k < x < k+1$

Math Question Bank