<p>A function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}\left( x \right)=\frac{2}{1+{{x}^{2}}}$ if $x\le 0$</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760688866858-hr5kd8x58um.png" width="400" class="aligncenter" /></figure>
<ol class="abc-list"><li><div class="list-item-paragraph">Give a reason why $\mathrm{g}$ does not have an inverse.</div><p style="text-align: right;">[1]</p></li><li><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 $k$.</div><p style="text-align: right;">[1]</p></li></ol>
<p>In the rest of the question, the domain of $\mathrm{g}$ is $x\le k$, where $k$ takes the value determined in part <strong>(b)</strong>.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find ${{\mathrm{g}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{g}}^{-1}}$.</div><p style="text-align: right;">[3]</p></li><li><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 $y={{\mathrm{g}}^{-1}}\left( x \right)$.</div><p style="text-align: right;">[3]</p><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760688937735-02l045a1yxko.png" alt="List item image" /></div></li></ol>

$\mathrm{g}\left( x \right)=\frac{2}{1+{{x}^{2}}}$ if $x\le 0$

Give a reason why $\mathrm{g}$ does not have an inverse.

The function ${{\mathrm{g}}^{-1}}$ exists if the domain of $\mathrm{g}$ is restricted to $x\le k$. State the greatest possible value of $k$.

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

Find ${{\mathrm{g}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{g}}^{-1}}$.

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 $y={{\mathrm{g}}^{-1}}\left( x \right)$.

2023 EJC Promo Q7

2023 EJC Promo Q7

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