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 as follows.<br>\[\mathrm{f}:x\mapsto 1+4x-x^2,\quad x&lt;a.\]</p>
<p>You are given that&nbsp;$\mathrm{f}^{-1}$ exists.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-3-item-0" data-list-item-path="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-3-item-0"><div class="list-item-paragraph">By the method of completing the square, state the largest possible value of $a$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For the rest of the question, take $a=0$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-5-item-0" data-list-item-path="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-5-item-0"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{f}^{-1}(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-5-item-1" data-list-item-path="h2-2025-rvhs-promo-q2-rvhs-q2-item-block-5-item-1"><div class="list-item-paragraph">Find the exact solution of the equation $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$.<span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined as follows.
\[\mathrm{f}:x\mapsto 1+4x-x^2,\quad x<a.\]

You are given that $\mathrm{f}^{-1}$ exists.

By the method of completing the square, state the largest possible value of $a$.

For the rest of the question, take $a=0$.

Find an expression for $\mathrm{f}^{-1}(x)$.

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

Intersections of a Function and Its Inverse

Restricting a Domain to Obtain an Inverse

Inverse Functions

2025 RVHS Promo Q2

Functions

<p>The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \dfrac{4x-1}{x+1},\ x&gt;-1$.</p>
<ol class="i-list"><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-0" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{f}$ has an inverse.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-1" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-1"><div class="list-item-paragraph">Find $\mathrm{f}^{-1}(x)$ and state its domain.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-2" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-2"><div class="list-item-paragraph">State the range of $\mathrm{f}^{-1}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-3" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-3"><div class="list-item-paragraph">State the equation of the line in which the graph of $y=\mathrm{f}(x)$ must be reflected in order to obtain the graph of $y=\mathrm{f}^{-1}(x)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-4" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-4"><div class="list-item-paragraph">Hence, or otherwise, find the exact solutions of the equation $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \dfrac{4x-1}{x+1},\ x>-1$.

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

State the equation of the line in which the graph of $y=\mathrm{f}(x)$ must be reflected in order to obtain the graph of $y=\mathrm{f}^{-1}(x)$.

Hence, or otherwise, find the exact solutions of the equation $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$.

Horizontal Line Test

2025 MI PU1 Promo Q9

<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.

Sketching 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 $g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$.

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1MzE4MzEzOTktcnZ6a3AyOGwzOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0OTA4LCJleHAiOjE3OTM1MDUyMDh9.PbALbtz67KvXYEowzKYCFS6lCEO_HZYI5A_HabyoQiI" 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</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 $k$.

Composite Functions

One-to-One Functions and Inverse Existence

Existence of Composite Functions

Range of a Composite Function

Iterates $f^n$ for Large $n$

2014 VJC Promo Q8

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

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

Piecewise-Defined Functions

2024 DHS P2 Q4

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \left| 4+2x-{{x}^{2}} \right|$, $x\in \mathbb{R}$, $x\ge 3.5$,</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto 4+\mathrm{e}{{\,}^{ax}}$, $x\in \mathbb{R}$, $x\ge -1$,</p>
<p>where&nbsp;$a &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760705342915-4q1mwpkpo" data-list-item-path="item-1760705342915-4q1mwpkpo"><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-1760705344001-gbw2x71kk" data-list-item-path="item-1760705344001-gbw2x71kk"><div class="list-item-paragraph">Find the value of $x$ for which ${{\mathrm{f}}^{-1}}\left( x \right)=\mathrm{f}\left( x \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760705344556-n8sgqyp9t" data-list-item-path="item-1760705344556-n8sgqyp9t"><div class="list-item-paragraph">Show that the composite function $\mathrm{fg}$ exists and express the exact range of $\mathrm{fg}$ in the form of $A+B{{\mathrm{e}}^{-a}}+C{{\mathrm{e}}^{-2a}}$, where&nbsp;$A$, $B$ and $C$ are real constants.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760705345418-qup0bymtl" data-list-item-path="item-1760705345418-qup0bymtl"><div class="list-item-paragraph">Without the use of a graphing calculator, solve the inequality $\frac{\mathrm{g}\left( x \right)}{{{x}^{2}}-2x-2}\ge 0$.</div><div class="list-item-paragraph">Leave your answer in exact form.<span class="inline-marks">[3]</span></div></li></ol>

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

$\mathrm{f}:x\mapsto \left| 4+2x-{{x}^{2}} \right|$, $x\in \mathbb{R}$, $x\ge 3.5$,

$\mathrm{g}:x\mapsto 4+\mathrm{e}{{\,}^{ax}}$, $x\in \mathbb{R}$, $x\ge -1$,

Modulus Functions

Inequalities involving Rational Functions

2024 ASRJC P1 Q8

Inequalities

<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 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 $0 < a < 1$.

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

2023 ASRJC 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

Range of a Function

Asymptotes and Discontinuities

2013 HCI C2 BT2 P1 Q11 [Modified]

Graphing Techniques

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

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

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

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

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

2018 MI Promo Q10

<ol class="abc-list"><li data-list-item-id="item-1760100909528-bp996gbrt" data-list-item-path="item-1760100909528-bp996gbrt"><div class="list-item-paragraph">The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto 1+\sqrt{x}$, for&nbsp;$x\in \mathbb{R}$, $x\ge 0$.</div><ol class="i-list"><li data-list-item-id="subitem-1760100921926-1biawe8az" data-list-item-path="item-1760100909528-bp996gbrt/subitem-1760100921926-1biawe8az"><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 data-list-item-id="subitem-1760100929133-qheo033mx" data-list-item-path="item-1760100909528-bp996gbrt/subitem-1760100929133-qheo033mx"><div class="list-item-paragraph">Show that if&nbsp;$\mathrm{ff}\left( x \right)=x$ then&nbsp;${{x}^{3}}-4{{x}^{2}}+4x-1=0$. Hence find the value of $x$ for which $\mathrm{ff}\left( x \right)=x$. Explain why this value of $x$ satisfies the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.<span class="inline-marks">[5]</span></div></li></ol></li><li data-list-item-id="item-1760100943946-c5l3ly5im" data-list-item-path="item-1760100943946-c5l3ly5im"><div class="list-item-paragraph">The function $\mathrm{g}$, with domain the set of non-negative integers, is given by</div><div class="list-item-paragraph" style="text-align: center"><p>${\mathrm{g}(n) = \begin{cases} 1 &amp; \mathrm{for } \, n = 0, \\ 2 + \mathrm{g}\left(\frac{1}{2} n\right) &amp; \mathrm{for } \,  n \, \mathrm{ even}, \\ 1 + \mathrm{g}(n - 1) &amp; \mathrm{for } \, n \, \mathrm{ odd}. \end{cases}}$</p></div><ol class="i-list"><li data-list-item-id="subitem-1760100972655-jq78suk4a" data-list-item-path="item-1760100943946-c5l3ly5im/subitem-1760100972655-jq78suk4a"><div class="list-item-paragraph">Find $\mathrm{g}\left( 4 \right)$, $\mathrm{g}\left( 7 \right)$and $\mathrm{g}\left( 12 \right)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760100973571-b9tkib0a1" data-list-item-path="item-1760100943946-c5l3ly5im/subitem-1760100973571-b9tkib0a1"><div class="list-item-paragraph">Does $\mathrm{g}$ have an inverse? Justify your answer.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

N2016 P1 Q10

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto {{x}^{2}}-4x-5$, for&nbsp;$x\in \mathbb{R}$, $x\le 2$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">On the same diagram, sketch the graphs of $\mathrm{f}$ and ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the exact solution of the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto 1+{{\mathrm{e}}^{2x}}$, for&nbsp;$x\in \mathbb{R}$, $x&lt;0$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Explain why the composite function $\mathrm{fg}$ exists and find the range of $\mathrm{fg}$.<span class="inline-marks">[3]</span></div></li></ol>

$\mathrm{f}:x\mapsto {{x}^{2}}-4x-5$, for $x\in \mathbb{R}$, $x\le 2$.

$\mathrm{g}:x\mapsto 1+{{\mathrm{e}}^{2x}}$, for $x\in \mathbb{R}$, $x<0$.

2022 YIJC Promo Q9

<p>The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \left| 3+x-2{{x}^{2}} \right|$ for $x\in \mathbb{R}$, $x\le k$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the greatest value of $k$ such that $\mathrm{f}$ has an inverse.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Using the value of $k$ in part (i), find the inverse function ${{\mathrm{f}}^{-1}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Sketch both $\mathrm{f}$ and ${{\mathrm{f}}^{-1}}$ on the same diagram. Is there any solution for $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$?</div><p style="text-align: right;">[3]</p></li></ol>
<p>Another function $\mathrm{g}$ is given by $\mathrm{g}:x\mapsto {{\mathrm{e}}^{-x}}$, $x\in \mathbb{R}$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Show that ${{\mathrm{f}}^{-1}}\mathrm{g}$ exists and find the range of this composite function.</div><p style="text-align: right;">[3]</p></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \left| 3+x-2{{x}^{2}} \right|$ for $x\in \mathbb{R}$, $x\le k$.

Another function $\mathrm{g}$ is given by $\mathrm{g}:x\mapsto {{\mathrm{e}}^{-x}}$, $x\in \mathbb{R}$.

ASRJC Functions Tutorial Q9

<p>Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto 2\left| x-p \right|+1$ for&nbsp;$x\in \mathbb{R}$, $p&gt;1$,</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto x\left( x-q \right)$ for&nbsp;$x\in \mathbb{R}$, $q&lt;0$.</p>
<ol class="i-list"><li data-list-item-id="item-1759915336582-t29efnmhv" data-list-item-path="item-1759915336582-t29efnmhv"><div class="list-item-paragraph">Explain why $\mathrm{f}$ does not have an inverse.<span class="inline-marks">[2]</span></div></li></ol>
<p>The domain of $\mathrm{f}$ is now restricted to $x\le k$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759915350309-wffem4k43" data-list-item-path="item-1759915350309-wffem4k43"><div class="list-item-paragraph">Write down the largest value of $k$ for which the function ${{\mathrm{f}}^{-1}}$ exists. Hence find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759915356944-76jexobik" data-list-item-path="item-1759915356944-76jexobik"><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. Hence solve $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759915358932-cx8xz7nh8" data-list-item-path="item-1759915358932-cx8xz7nh8"><div class="list-item-paragraph">Find the range of $\mathrm{g}{{\mathrm{f}}^{-1}}$.<span class="inline-marks">[2]</span></div></li></ol>

$\mathrm{f}:x\mapsto 2\left| x-p \right|+1$ for $x\in \mathbb{R}$, $p>1$,

$\mathrm{g}:x\mapsto x\left( x-q \right)$ for $x\in \mathbb{R}$, $q<0$.

The domain of $\mathrm{f}$ is now restricted to $x\le k$.

2019 NJC P1 Q9

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \left( x+3 \right)\left| x-2 \right|$, </p>
<p style="text-align: center">$x\in \mathbb{R}$, $\mathrm{g}:x\mapsto \frac{24}{{{x}^{2}}+3}-1$, $x\in \mathbb{R}$, $x>0$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain why the function ${{\mathrm{f}}^{-1}}$ does not exist.</div><p style="text-align: right;">[1]</p></li></ol>
<p>In the rest of the question, the domain of $\mathrm{f}$ is restricted to $\left[ k,2 \right]$ where $k$ is the least value such that ${{\mathrm{f}}^{-1}}$ exists.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Write down the value of $k$. Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Sketch the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$on the same diagram, showing clearly the relationship between the two graphs. Hence, find the exact solution of the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Determine whether ${{\mathrm{g}}^{-1}}\mathrm{f}$ exists.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that $\mathrm{g}$ is a strictly decreasing function. Hence, without finding ${{\mathrm{g}}^{-1}}$, find the range of values of $x$ for which ${{\mathrm{g}}^{-1}}\left( \mathrm{f}\left( x \right) \right)<1$, giving your answer in exact form.</div><p style="text-align: right;">[3]</p></li></ol>

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

$x\in \mathbb{R}$, $\mathrm{g}:x\mapsto \frac{24}{{{x}^{2}}+3}-1$, $x\in \mathbb{R}$, $x>0$.

In the rest of the question, the domain of $\mathrm{f}$ is restricted to $\left[ k,2 \right]$ where $k$ is the least value such that ${{\mathrm{f}}^{-1}}$ exists.

Math Question Bank