A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

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

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

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

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

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

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

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

Composite Functions

Sketching a Function and Its Inverse

Existence of Composite Functions

Inverse Trigonometric Functions

Restricting a Domain to Obtain an Inverse

Range of a Composite Function

Inverse Functions

Inverting a shifted sine and forming a reciprocal composite

Functions

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f}: x \mapsto x^2 - 4x + 5, x \in \mathbb{R}, x &gt; 2.\]</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph">Find $\mathrm{f}^{-1}(x)$ and write down the domain of $\mathrm{f}^{-1}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Sketch the graph of $\mathrm{f}$ and $\mathrm{f}^{-1}$ on the same diagram.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-2-item-2" data-list-item-path="imported-item-block-2-item-2"><div class="list-item-paragraph">Explain why the $x$-coordinate of the point of intersection of the curves in part <strong>(b)</strong> satisfies the equation</div><div class="list-item-paragraph" style="text-align: center">\[x^2 - 5x + 5 = 0\]</div><div class="list-item-paragraph">and find the exact value of this $x$-coordinate.<span class="inline-marks">[3]</span></div></li></ol>

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

\[\mathrm{f}: x \mapsto x^2 - 4x + 5, x \in \mathbb{R}, x > 2.\]

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

Sketch the graph of $\mathrm{f}$ and $\mathrm{f}^{-1}$ on the same diagram.

Intersections of a Function and Its Inverse

<p>Consider a function $g$ defined as&nbsp;$g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$.</p>
<ol class="abc-list"><li data-list-item-id="inverse-restrict-q3a" data-list-item-path="inverse-restrict-q3a"><div class="list-item-paragraph">Sketch the graph of $g$ and hence explain why $g$ cannot have an inverse.</div></li><li data-list-item-id="inverse-restrict-q3b" data-list-item-path="inverse-restrict-q3b"><div class="list-item-paragraph">If&nbsp;$g:x\mapsto (x+2)^2-3,\ x\ge q$ has an inverse, write down the smallest value $q$ can take.</div></li><li data-list-item-id="inverse-restrict-q3c" data-list-item-path="inverse-restrict-q3c"><div class="list-item-paragraph">For this value of $q$, find the function $g^{-1}$.</div></li><li data-list-item-id="inverse-restrict-q3d" data-list-item-path="inverse-restrict-q3d"><div class="list-item-paragraph">Sketch the graphs of $g$ and $g^{-1}$ on the same axes, clearly showing the relationship between them.</div></li><li data-list-item-id="inverse-restrict-q3e" data-list-item-path="inverse-restrict-q3e"><div class="list-item-paragraph">Write down the range of $g^{-1}$.</div></li><li data-list-item-id="item-1783676614199-a1m0ed6xb" data-list-item-path="item-1783676614199-a1m0ed6xb"><div class="list-item-paragraph">Find the exact solution of the equation $g (x) =g^{-1}(x)$.</div></li></ol>

Consider a function $g$ defined as $g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$.

Horizontal Line Test

<p>The curve with equation $y = \mathrm{f}(x)$ for&nbsp;$x \leq b$, where&nbsp;$\mathrm{f}(x)$ is a <strong>quadratic</strong> function, intersects the $x$-axis at the points $(a,0)$ and $(b,0)$, and the $y$-axis at the point $(0,c)$. The graph of $y = \mathrm{f}(x)$ is shown in Figure 1. The scales on the $x$- and $y$-axes are the same.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1783531831399-rvzkp28l39.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1MzE4MzEzOTktcnZ6a3AyOGwzOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0OTE0LCJleHAiOjE3OTM1MDUyMTR9.ke_KkByFKqrWJ5V-2j6ZmlxmvMiIsWCEKhfcQUx0RBM" style="max-width: min(300px, 100%); height: auto; display: block;" alt="Figure 1" class="aligncenter custom-width" /></figure><div class="image-caption" style="text-align: center;">Figure 1</div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><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">Sketch the graph of $y = \mathrm{f}(\left| x \right|)$, labelling the points where the graph intersects or touches the axes.<span class="inline-marks">[2]</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">Sketch the graph of $y = \frac{1}{\mathrm{f}(x)}$, labelling the points where the graph intersects or touches the axes, as well as the equations of any asymptotes.<span class="inline-marks">[2]</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">Describe fully a sequence of transformations which transforms the graph of $y = \mathrm{f}(2x+1)$ onto the graph of $y = \mathrm{f}(x)$.<span class="inline-marks">[2]</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 $y = \mathrm{g}(x)$ is such that&nbsp;$\mathrm{g}(x) = \mathrm{f}(x)$ for&nbsp;$x \leq k$, and $y = \mathrm{g}^{-1}(x)$ exists.</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">Find the largest possible value of $k$ in terms of $a$ and $b$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">Using the value of $k$ found in <strong>(b)(i)</strong>, sketch the graph of $y = \mathrm{g}^{-1}(x)$ on Figure 1 in the Answer Booklet, labelling the intersections with the axes. Write down the range of $\mathrm{g}^{-1}$ in terms of $a$ and $b$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-iii" data-list-item-path="item-b/item-b-iii"><div class="list-item-paragraph">Explain why the solution to $\mathrm{g}^{-1}(x) = x$ satisfies the equation $\mathrm{g}^{-1}(x) = \mathrm{g}(x)$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

The curve with equation $y = \mathrm{f}(x)$ for $x \leq b$, where $\mathrm{f}(x)$ is a **quadratic** function, intersects the $x$-axis at the points $(a,0)$ and $(b,0)$, and the $y$-axis at the point $(0,c)$. The graph of $y = \mathrm{f}(x)$ is shown in Figure 1. The scales on the $x$- and $y$-axes are the same.

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

Graph Transformations of $f(ax+b)$

Sketching Graphs of $f(\lvert x\rvert)$

2025 ACJC P1 Q8

Transformations of Graphs

<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 $\mathrm{f} : x \mapsto 1 - (x + 3)^2$, $x \in \mathbb{R}$, $x \ge -2$.

One-to-One Functions and Inverse Existence

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

2011 NJC P1 Q7

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

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

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

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

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

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

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

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

Modulus Functions

2024 DHS Promo Q10

<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.

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>

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

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

Recovering a Function from a Composite

2025 DHS P1 Q7

Applications of Differentiation

<p>The function $g$ is defined by
$g: x \mapsto x^{2} + 2x - 3,\ x \in \mathbb{R},\ x > -1$.

Sketch the graphs of $y = \mathrm{g}(x)$ and $y = g^{-1}(x)$, in</p>

The function $g$ is defined by
$g: x \mapsto x^{2} + 2x - 3,\ x \in \mathbb{R},\ x > -1$.

Sketch the graphs of $y = \mathrm{g}(x)$ and $y = g^{-1}(x)$, in

How to Use GC to Plot an Inverse Graph

<p>The function $g$ is defined by $g: x \mapsto x^{2} + 2x - 3,  x \in \mathbb{R}, x &gt; -1$. Sketch the graphs of $y = \mathrm{g}(x)$ and $y = g^{-1}(x)$, indicating the relationship of the two graphs, and find the exact value of $x$ for which $\mathrm{g}(x) = g^{-1}(x)$.</p>

The function $g$ is defined by $g: x \mapsto x^{2} + 2x - 3,  x \in \mathbb{R}, x > -1$. Sketch the graphs of $y = \mathrm{g}(x)$ and $y = g^{-1}(x)$, indicating the relationship of the two graphs, and find the exact value of $x$ for which $\mathrm{g}(x) = g^{-1}(x)$.

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}\left( x \right)=1+\sqrt{x}$, $x\in \mathbb{R}$, $x\ge 0$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Evaluate $\mathrm{ff}\left( 9 \right)$.</div></li><li><div class="list-item-paragraph">Find an expression for the inverse function ${{\mathrm{f}}^{-1}}\left( x \right)$.</div></li><li><div class="list-item-paragraph">Sketch in the same diagram the graph of $\mathrm{f}\left( x \right)$ and the graph of ${{\mathrm{f}}^{-1}}\left( x \right)$, clearly marking the line of reflection between the two graphs.</div></li><li><div class="list-item-paragraph">Show that $x=\frac{3+\sqrt{5}}{2}$ is the only solution of the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.</div></li></ol>

$\mathrm{f}\left( x \right)=1+\sqrt{x}$, $x\in \mathbb{R}$, $x\ge 0$.

J2 A Level Revision Set: Functions Q1

<p>The function ${\mathrm{f}}$ is defined by</p>
<p style="text-align: center">${\mathrm{f}\left( x \right)=2+\sqrt{x}}$, ${x\in \mathbb{R}}$, ${x\ge 0}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760840803757-m48yb3k1z" data-list-item-path="item-1760840803757-m48yb3k1z"><div class="list-item-paragraph">Evaluate ${\mathrm{ff}\left( 49 \right)}$.</div></li><li data-list-item-id="item-1760840804334-kqqdkdzc6" data-list-item-path="item-1760840804334-kqqdkdzc6"><div class="list-item-paragraph">Find an expression for the inverse function, ${{{\mathrm{f}}^{-1}}\left( x \right)}$.</div></li><li data-list-item-id="item-1760840806397-ut16vee1c" data-list-item-path="item-1760840806397-ut16vee1c"><div class="list-item-paragraph">Sketch in the same set of axes of the graph of ${\mathrm{f}\left( x \right)}$ and the graph of ${{{\mathrm{f}}^{-1}}\left( x \right)}$, clearly marking the line of reflection between the two graphs.</div></li><li data-list-item-id="item-1760840807184-izmdmq7e8" data-list-item-path="item-1760840807184-izmdmq7e8"><div class="list-item-paragraph">Show that&nbsp;${x=4}$ is the only solution of the equation ${\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)}$.</div></li></ol>

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

${\mathrm{f}\left( x \right)=2+\sqrt{x}}$, ${x\in \mathbb{R}}$, ${x\ge 0}$.

Functions Revision Worksheet Q1

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{{{x}^{2}}}{x-4}$, $x\in \mathbb{R}$, $4 < x \le 8$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and write down the domain of ${{\mathrm{f}}^{-1}}$. On the same axes, sketch the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The region $R$ is bounded by the curve $y={{\mathrm{f}}^{-1}}\left( x \right)$, the lines $y=5$, $y=8$ and the $y$-axis. Find the exact area of $R$.</div><p style="text-align: right;">[3]</p></li></ol>

$\mathrm{f}:x\mapsto \frac{{{x}^{2}}}{x-4}$, $x\in \mathbb{R}$, $4 < x \le 8$

2024 DHS P1 Q5

<p>A function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">${\mathrm{g}(x) = \begin{cases} \frac{2}{1+x^2} & \text{if}\hspace{0.5em} x \leq 0, \\ 0 & \text{if}\hspace{0.5em} x &gt; 0. \end{cases}}$</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760688866858-hr5kd8x58um.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA2ODg4NjY4NTgtaHI1a2Q4eDU4dW0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDkxNCwiZXhwIjoxNzkzNTA1MjE0fQ.L3khEH3UfzZfMuQpAkE3TP9NEpvblkeGVNHmK3ODeeo" style="max-width: min(450px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1760688876762-r45qipthu" data-list-item-path="item-1760688876762-r45qipthu"><div class="list-item-paragraph">Give a reason why $\mathrm{g}$ does not have an inverse.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760688877385-kldko7ckx" data-list-item-path="item-1760688877385-kldko7ckx"><div class="list-item-paragraph">The function ${{\mathrm{g}}^{-1}}$ exists if the domain of $\mathrm{g}$ is restricted to $x\le k$. State the greatest possible value of $k$.<span class="inline-marks">[1]</span></div></li></ol>
<p>In the rest of the question, the domain of $\mathrm{g}$ is $x\le k$, where&nbsp;$k$ takes the value determined in part <strong>(b)</strong>.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760688906709-nqijnxhe4" data-list-item-path="item-1760688906709-nqijnxhe4"><div class="list-item-paragraph">Find ${{\mathrm{g}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{g}}^{-1}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760688907760-py1k7wfly" data-list-item-path="item-1760688907760-py1k7wfly"><div class="list-item-paragraph">Sketch, on the axes given below, the graphs of $y=\mathrm{g}\left( x \right)$ and $y={{\mathrm{g}}^{-1}}\left( x \right)$. Label the two graphs clearly. Write down the equation of the line in which the graph of $y=\mathrm{g}\left( x \right)$ must be reflected in order to obtain the graph of $y={{\mathrm{g}}^{-1}}\left( x \right)$.<span class="inline-marks">[3]</span></div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760688937735-02l045a1yxko.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA2ODg5Mzc3MzUtMDJsMDQ1YTF5eGtvLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTM1MDQ5MTQsImV4cCI6MTc5MzUwNTIxNH0.TxnQczQDaZzAtkQf2leEo-07UYQtygiPczhWnfTVH4s" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li></ol>

${\mathrm{g}(x) = \begin{cases} \frac{2}{1+x^2} & \text{if } x \leq 0, \\ 0 & \text{if } x > 0. \end{cases}}$

In the rest of the question, the domain of $\mathrm{g}$ is $x\le k$, where $k$ takes the value determined in part **(b)**.

2023 EJC Promo Q7

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{1}{{{x}^{2}}-4ax+5{{a}^{2}}}$, $x\in \mathbb{R}$, $x < k$,</p>
<p>where $a$ and $k$ are positive constants.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the largest possible value of $k$, in terms of $a$, such that ${{\mathrm{f}}^{-1}}$ exists.</div><p style="text-align: right;">[2]</p></li></ol>
<p>It is now given that $k=a$ and $a > 1$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Show that the composite function ${{\mathrm{f}}^{2}}$ exists.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ in terms of $x$ and $a$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">On the same diagram, sketch the graphs of $\mathrm{f}$ and ${{\mathrm{f}}^{-1}}$, giving the equations of any asymptotes.</div><p style="text-align: right;">[4]</p></li></ol>

$\mathrm{f}:x\mapsto \frac{1}{{{x}^{2}}-4ax+5{{a}^{2}}}$, $x\in \mathbb{R}$, $x < k$,

where $a$ and $k$ are positive constants.

Sketching Rational Functions

Asymptotes and Discontinuities

2023 YIJC P1 Q8

Graphing Techniques

<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">$\mathrm{f}:x\mapsto {{\left( x-2 \right)}^{2}}-3$, $x\le 2$</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto 4\cos x$, $0\le x<\pi $</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}$ in a similar form.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Sketch on the same diagram, the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$.</div><div class="list-item-paragraph">Hence, solve the equation ${{\mathrm{f}}^{-1}}\left( x \right)=\mathrm{f}\left( x \right)$, leaving your answer in exact form.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Without finding ${{\mathrm{g}}^{-1}}$, solve ${{\mathrm{g}}^{-1}}\left( x \right)=\frac{\pi }{4}$, giving your answer in exact form.</div><p style="text-align: right;">[2]</p></li></ol>

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

$\mathrm{f}:x\mapsto {{\left( x-2 \right)}^{2}}-3$, $x\le 2$

$\mathrm{g}:x\mapsto 4\cos x$, $0\le x<\pi $

2008 YJC P1 Q6 [Modified]

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \ln \left( {{x}^{2}}-6x+10 \right)-5$ , $x\in \mathbb{R}$, $x\le 3$.</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 on the same diagram the graphs of $y=\mathrm{f}\left( x \right)$ and $y=\mathrm{f}{{\,}^{-1}}\left( x \right)$, giving the coordinates of the axial intercepts.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \frac{1}{x-5}+x-2$, $x\in \mathbb{R}$, $x\ne 5$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the range of $\mathrm{g}{{\mathrm{f}}^{-1}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Solve the inequality $\mathrm{f}\left( x \right)&lt;\mathrm{g}\left( x \right)$.<span class="inline-marks">[2]</span></div></li></ol>

$\mathrm{f}:x\mapsto \ln \left( {{x}^{2}}-6x+10 \right)-5$ , $x\in \mathbb{R}$, $x\le 3$.

$\mathrm{g}:x\mapsto \frac{1}{x-5}+x-2$, $x\in \mathbb{R}$, $x\ne 5$.

Graphical Method

2022 RI Promo Q10

Inequalities

<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>

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

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

2023 ASRJC P2 Q4

<p>The graph below shows the graph of $y=\mathrm{f}\left( x \right)$.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760352383525-10uyylq9iz1m.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzNTIzODM1MjUtMTB1eXlscTlpejFtLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTM1MDQ5MTQsImV4cCI6MTc5MzUwNTIxNH0.LnNizmP18H48yLu4_Vsykxi44tDeHMdaw9rfvBPidzQ" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch on the same diagram the graph of $y={{\mathrm{f}}^{-1}}\left( x \right)$, clearly stating the equations of any asymptotes and the coordinates of the points where the curve crosses the axes. State also the geometrical relationship between these two graphs.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{x}^{2}}+6x+5$, $x\in \mathbb{R}$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the range of the composite function $\mathrm{fg}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">It is given that $\mathrm{fg}\left( a \right)=0$. Find the value of $\mathrm{g}\left( a \right)$, where $a$ is a negative constant.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The function $\mathrm{h}$ is defined by $\mathrm{h}:x\mapsto {{x}^{2}}+6x+5$, $x\in \mathbb{R}$, $x<-4$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find ${{\mathrm{h}}^{-1}}$ and state the domain of ${{\mathrm{h}}^{-1}}$.</div><p style="text-align: right;">[3]</p></li></ol>

The graph below shows the graph of $y=\mathrm{f}\left( x \right)$.

The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{x}^{2}}+6x+5$, $x\in \mathbb{R}$.

The function $\mathrm{h}$ is defined by $\mathrm{h}:x\mapsto {{x}^{2}}+6x+5$, $x\in \mathbb{R}$, $x<-4$.

Math Question Bank