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>Given&nbsp;$h(x)=1-|x-2|$, where&nbsp;$0\le x\le4$, and $H(x)=\dfrac{1}{h(x)}$, the range of $H(x)$ is the set&nbsp;$R$ such that&nbsp;$R=\{y:y\ge a\text{ or}\hspace{0.5em}y\le b\}$ where&nbsp;$a,b\in\mathbb Z$.</p>
<p>Determine the values of $a$ and $b$.</p>

Given $h(x)=1-|x-2|$, where $0\le x\le4$, and $H(x)=\dfrac{1}{h(x)}$, the range of $H(x)$ is the set $R$ such that $R=\{y:y\ge a\text{ or }y\le b\}$ where $a,b\in\mathbb Z$.

Modulus Functions

Range of a Function

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

Graphs of Modulus Function

Two branches in a reciprocal modulus range

Functions

Modulus Function

Transformations of Graphs

<p>For&nbsp;$-2\le x\le3$, define $p(x)=|2x-1|-3$ and $q(x)=\dfrac{1}{p(x)}$ wherever $p(x)\ne0$. The range of $q$ is $\{y:y\le a\text{ or}\hspace{0.5em}y\ge b\}$. Find the exact values of $a$ and $b$.<span class="inline-marks">[4]</span></p>

For $-2\le x\le3$, define $p(x)=|2x-1|-3$ and $q(x)=\dfrac{1}{p(x)}$ wherever $p(x)\ne0$. The range of $q$ is $\{y:y\le a\text{ or }y\ge b\}$. Find the exact values of $a$ and $b$.

Reciprocal range of a stretched modulus trough

<p>The function $\mathrm{f}$ is defined by \[\mathrm{f}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\pi\leq x\leq\pi.\]</p>
<ol class="abc-list"><li data-list-item-id="rf-33" data-list-item-path="rf-33"><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the form $R\cos(x+\alpha)$, 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-34" data-list-item-path="rf-34"><div class="list-item-paragraph">Hence state the range of $\mathrm{f}$ and sketch the graph of $y=\mathrm{f}(x)$, indicating the coordinates of the turning points and the intersections with the coordinate axes.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by \[\mathrm{g}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\alpha\leq x\leq b,\] where&nbsp;$b&gt;-\alpha$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="rf-37" data-list-item-path="rf-37"><div class="list-item-paragraph">Find the largest value of $b$ for which $\mathrm{g}^{-1}$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rf-38" data-list-item-path="rf-38"><div class="list-item-paragraph">Using this value of $b$, find $\mathrm{g}^{-1}(x)$, stating its domain and range.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by \[\mathrm{f}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\pi\leq x\leq\pi.\]

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

Hence state the range of $\mathrm{f}$ and sketch the graph of $y=\mathrm{f}(x)$, indicating the coordinates of the turning points and the intersections with the coordinate axes.

The function $\mathrm{g}$ is defined by \[\mathrm{g}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\alpha\leq x\leq b,\] where $b>-\alpha$.

Find the largest value of $b$ for which $\mathrm{g}^{-1}$ exists.

Using this value of $b$, find $\mathrm{g}^{-1}(x)$, stating its domain and range.

Inverse Trigonometric Functions

Restricting a Domain to Obtain an Inverse

Inverse Functions

Restricting a shifted cosine for an inverse

<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 Functions

Composite Transformations

Existence of Composite Functions

Range of a Composite Function

2025 MI PU1 Promo Q7 [Modified]

<p>The variables $x$ and $y$ satisfy the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=y^3-\frac{y}{x}$, where&nbsp;$x,y&gt;0$. It is given that&nbsp;$y=2$ when&nbsp;$x=1$.</p>
<p>Euler's method is a numerical method for estimating a solution of a differential equation. Over a small horizontal step of length $h$, it uses a straight line with the gradient given by the differential equation at the current estimated point.</p>
<p>The estimated change in $y$ is therefore&nbsp;$h\times\text{current gradient}$. Let&nbsp;$y_n$ be the estimate of $y$ at $x=x_n$, with $x_0=1$ and $y_0=2$. For this differential equation, the rule is</p>
<p>\[x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h\left(y_n^3-\frac{y_n}{x_n}\right).\]</p>
<p>At each step, calculate the gradient using the current values of $x$ and $y$, then add $h$ times this gradient to the current estimate of $y$. Recalculate the gradient at the new estimated point before repeating. Keep unrounded values in subsequent calculations.</p>
<ol class="abc-list"><li data-list-item-id="p2-tzc-q12-item01" data-list-item-path="p2-tzc-q12-item01"><div class="list-item-paragraph">Take $h=0.01$. First calculate the gradient at $(1,2)$. Use this gradient in the Euler formula to show that the first estimate, at $x_1=1.01$, is $y_1=2.06$.</div><div class="list-item-paragraph">Then calculate $y_2$, showing your substitution and giving its value to 5 decimal places. Continue the process to estimate $y$ when&nbsp;$x=1.05$, giving this estimate to 3 significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item02" data-list-item-path="p2-tzc-q12-item02"><div class="list-item-paragraph">Using the substitution $u=xy$, show that the differential equation can be written as&nbsp;$\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{u^3}{x^2}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item03" data-list-item-path="p2-tzc-q12-item03"><div class="list-item-paragraph">Hence, by separating the variables and using the given initial condition, solve the differential equation, giving $y$ as a function of $x$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="p2-tzc-q12-item04" data-list-item-path="p2-tzc-q12-item04"><div class="list-item-paragraph">State the largest possible domain of this solution containing $x=1$, and its corresponding range.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item05" data-list-item-path="p2-tzc-q12-item05"><div class="list-item-paragraph">By considering the shape of the solution curve for&nbsp;$1\leq x\leq1.05$ and the straight-line steps in Euler's method, state whether the approximation in part (a) is an overestimate or an underestimate. Give a reason.<span class="inline-marks">[1]</span></div></li></ol>

The variables $x$ and $y$ satisfy the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=y^3-\frac{y}{x}$, where $x,y>0$. It is given that $y=2$ when $x=1$.

Euler's method is a numerical method for estimating a solution of a differential equation. Over a small horizontal step of length $h$, it uses a straight line with the gradient given by the differential equation at the current estimated point.

The estimated change in $y$ is therefore $h\times\text{current gradient}$. Let $y_n$ be the estimate of $y$ at $x=x_n$, with $x_0=1$ and $y_0=2$. For this differential equation, the rule is

\[x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h\left(y_n^3-\frac{y_n}{x_n}\right).\]

At each step, calculate the gradient using the current values of $x$ and $y$, then add $h$ times this gradient to the current estimate of $y$. Recalculate the gradient at the new estimated point before repeating. Keep unrounded values in subsequent calculations.

Euler's Method

Concavity of Graphs

Solving Differential Equations by Separation of Variables

Function Notation, Domain and Range

Solving Differential Equations by Substitution

Product Rule

Domain of a Function

Sketching Solution Curves of Differential Equations

IB AA HL May 2026 P2 TZC Q12 [modified]

Differential Equations

Techniques of Differentiation

Applications of Differentiation

<p>The function $y=\mathrm{f}(x)$ is defined by $\mathrm{f}(x)=x^2-2x+6$, with $x\leq1$.</p>
<ol class="abc-list"><li data-list-item-id="p1q4-a" data-list-item-path="p1q4-a"><div class="list-item-paragraph">Find $\mathrm{f}^{-1}(x)$ and state its domain.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p1q4-b" data-list-item-path="p1q4-b"><div class="list-item-paragraph">Find the range of the function $\mathrm{g}(x)=3+\sqrt{\mathrm{f}(x)}$.<span class="inline-marks">[2]</span></div></li></ol>

The function $y=\mathrm{f}(x)$ is defined by $\mathrm{f}(x)=x^2-2x+6$, with $x\leq1$.

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

Find the range of the function $\mathrm{g}(x)=3+\sqrt{\mathrm{f}(x)}$.

IB Math HL January 2016 P1 Q4

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto x - 2 \mathrm{tan}^{-1}(2x)$, $-\frac{\sqrt{3}}{2} &lt; x &lt; \frac{\sqrt{3}}{2}$.</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{f}(x)$ decreases as&nbsp;$x$ increases.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Find the range of $\mathrm{f}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-iii" data-list-item-path="item-a/item-a-iii"><div class="list-item-paragraph">Explain whether the composite function $\mathrm{ff}$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-iv" data-list-item-path="item-a/item-a-iv"><div class="list-item-paragraph">The curve $C$ is obtained by reflecting the graph of $\mathrm{f}$ in the line $y = x$. Write down the equation of $C$.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The function $\mathrm{h}$ is defined by $\mathrm{h} : x \mapsto x^2 + 2$, $x \in \mathbb{R}$. The function $\mathrm{g}$ is such that the composite function $\mathrm{gh}$ exists and \[\mathrm{gh}(x) = x^4 - 2x^2 + 5, \quad x \in \mathbb{R}.\] Find an expression for&nbsp;$\mathrm{g}(x)$.<span class="inline-marks">[2]</span></div></li></ol>

Recovering a Function from a Composite

2009 IJC P1 Q10

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \rightarrow \left|\frac{2x + 4}{3 - x}\right|, \quad x \in \mathbb{R}, \quad x \neq 3.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = \mathrm{f}(x)$, giving the equations of any asymptotes and the coordinates of the points where the curve meets the axes.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence state the range of $\mathrm{f}$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain why the function $\mathrm{f}^2$ does not exist.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The domain of $\mathrm{f}$ is further restricted to $x \leqslant a$, where&nbsp;$a$ is a constant.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">State the greatest value of $a$ such that the function $\mathrm{f}^{-1}$ exists.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find $\mathrm{f}^{-1}(x)$ and state its domain.</span><span class="inline-marks">[4]</span></div></li></ol>

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

\[\mathrm{f} : x \rightarrow \left|\frac{2x + 4}{3 - x}\right|, \quad x \in \mathbb{R}, \quad x \neq 3.\]

Sketch the graph of $y = \mathrm{f}(x)$, giving the equations of any asymptotes and the coordinates of the points where the curve meets the axes.

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

The domain of $\mathrm{f}$ is further restricted to $x \leqslant a$, where $a$ is a constant.

State the greatest value of $a$ such that the function $\mathrm{f}^{-1}$ exists.

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

Sketching Rational Functions

Asymptotes and Discontinuities

N2023 P1 Q7

Graphing Techniques

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

Iterates $f^n$ for Large $n$

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

2025 HCI P1 Q8

<p>The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \frac{1}{x^2 + 6x + 5}$, for&nbsp;$x \ge -3$, $x \neq -1$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{f}^{-1}(x)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find algebraically the range of $\mathrm{f}$.</span><span class="inline-marks">[4]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto e^x$, for&nbsp;$x \in \mathbb{R}$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the exact range of $\mathrm{gf}$.</span><span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \frac{1}{x^2 + 6x + 5}$, for $x \ge -3$, $x \neq -1$.

Find algebraically the range of $\mathrm{f}$.

The function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto e^x$, for $x \in \mathbb{R}$.

2025 ACJC P2 Q4

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

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

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



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

One-to-One Functions and Inverse Existence

Piecewise-Defined Functions

Worked Example 3

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

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

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

<p>Find the exact range of each of the following functions:</p>
<p>$h:x\mapsto 2-{{e}^{x}},x\in \mathbb{R},x&lt;0$</p>

Find the exact range of each of the following functions:

$h:x\mapsto 2-{{e}^{x}},x\in \mathbb{R},x<0$

<p>Find the exact range of each of the following functions:</p>
<p>$g:x\mapsto {{x}^{2}}+3x-2,x\in \mathbb{R},x&lt;0$</p>

$g:x\mapsto {{x}^{2}}+3x-2,x\in \mathbb{R},x<0$

<p>Find the exact range of each of the following functions:</p>
<p>$f:x\mapsto \ln x,x\in \mathbb{R},x\ge 2$</p>

$f:x\mapsto \ln x,x\in \mathbb{R},x\ge 2$

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

Periodic Functions

2014 SRJC Promo Q11

<p>The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto \frac{4{{x}^{2}}-7x-1}{x-2}$, $x\in \mathbb{R},\ x\ne 2$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, indicating its turning points and asymptotes.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">State the range of $\mathrm{f}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The function $\mathrm{g}$, with domain the set of positive integers, is given by</p>
<p style="text-align: center">${\mathrm{g}(n) = \begin{cases} 
n, & \mathrm{for } n \mathrm{ odd} \\
2\mathrm{g}\left(\frac{n}{2}\right) + 1, & \mathrm{for } n \mathrm{ even} 
\end{cases}}$</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find $\mathrm{g}\left( 2 \right)$ and $\mathrm{g}\left( 8 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Does $\mathrm{fg}$ exist? Justify your answer.</div><p style="text-align: right;">[2]</p></li></ol>

The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto \frac{4{{x}^{2}}-7x-1}{x-2}$, $x\in \mathbb{R},\ x\ne 2$.

The function $\mathrm{g}$, with domain the set of positive integers, is given by

${\mathrm{g}(n) = \begin{cases} 
n, & \mathrm{for } n \mathrm{ odd} \\
2\mathrm{g}\left(\frac{n}{2}\right) + 1, & \mathrm{for } n \mathrm{ even} 
\end{cases}}$

Finding Possible y-values of a Curve

2022 EJC Promo Q10

<p>The functions ${\mathrm{f}}$ and ${\mathrm{g}}$ are defined by</p>
<p style="text-align: center">${\mathrm{f}:x\mapsto \sqrt{x-2}}$, ${x\in \mathbb{R}}$, ${x\ge 2}$</p>
<p style="text-align: center">${\mathrm{g}:x\mapsto 3{{x}^{2}}-6x+5}$, ${x\in \mathbb{R}}$, ${0\le x\le 4}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760868992904-2hyde0vdv" data-list-item-path="item-1760868992904-2hyde0vdv"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ and state the range of $\mathrm{f}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760868993465-u3bseo008" data-list-item-path="item-1760868993465-u3bseo008"><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and write down the domain of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760868994104-nudolr530" data-list-item-path="item-1760868994104-nudolr530"><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-1760868995454-8ffbkxyuw" data-list-item-path="item-1760868995454-8ffbkxyuw"><div class="list-item-paragraph">Find $\mathrm{fg}\left( x \right)$ and find the range of ${\mathrm{fg}}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The domain of $\mathrm{g}$ is now defined to be $x\in \mathbb{R}$.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1760869020266-86gmydwrt" data-list-item-path="item-1760869020266-86gmydwrt"><div class="list-item-paragraph">Describe fully a sequence of transformations which transforms the curve $y=\mathrm{g}\left( x \right)$ onto the curve ${y={{x}^{2}}-1}$.<span class="inline-marks">[3]</span></div></li></ol>

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

${\mathrm{f}:x\mapsto \sqrt{x-2}}$, ${x\in \mathbb{R}}$, ${x\ge 2}$

${\mathrm{g}:x\mapsto 3{{x}^{2}}-6x+5}$, ${x\in \mathbb{R}}$, ${0\le x\le 4}$.

The domain of $\mathrm{g}$ is now defined to be $x\in \mathbb{R}$.

2023 MI PRU2 P1 Q8

<p>The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \frac{2x}{x-2}$, for&nbsp;$x\in \mathbb{R}$, $x\ne 2$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760705458908-o8q06syho" data-list-item-path="item-1760705458908-o8q06syho"><div class="list-item-paragraph">Sketch the graph of $\mathrm{f}$ and find its range.<span class="inline-marks">[3]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto 3+\left| x+2 \right|$, for&nbsp;$x\in \mathbb{R}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760705473671-4snuh8zz9" data-list-item-path="item-1760705473671-4snuh8zz9"><div class="list-item-paragraph">Show that the composite function $\mathrm{fg}$ exists. Find $\mathrm{fg}(x)$ and state the domain and range of $\mathrm{fg}$.<span class="inline-marks">[5]</span></div></li></ol>

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

Another function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto 3+\left| x+2 \right|$, for $x\in \mathbb{R}$.

2024 RI P2 Q1

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

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

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

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