<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}(x)=\left\{ \begin{matrix}  \frac{x}{2}  \qquad \text{ if}\hspace{0.5em} x\le 0 \\
   2\sin x   \qquad  \text{ if}\hspace{0.5em}   0&lt;x\le 4 \\
\end{matrix} \right.$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}(x)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">If the domain of $\mathrm{f}$ is restricted to $x\le k$, state the largest value of $k$, in exact form, for which the function ${{\mathrm{f}}^{-1}}$ exist.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Using the domain from part <strong>(ii)</strong>, define ${{\mathrm{f}}^{-1}}$ in a similar form.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Solve ${{\mathrm{f}}^{-1}}(x)=\mathrm{f}(x)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>In the rest of the question, the domain of f is as originally defined.</p>
<p>The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto -{{x}^{3}},\mathrm{ }x\in \mathbb{R},\mathrm{ }x&gt;0$</p>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{fg}\left( x \right)$.<span class="inline-marks">[2]</span></div></li></ol>

$\mathrm{f}(x)=\left\{ \begin{matrix}  \frac{x}{2}  \qquad \text{ if } x\le 0 \\
   2\sin x   \qquad  \text{ if }   0<x\le 4 \\
\end{matrix} \right.$

If the domain of $\mathrm{f}$ is restricted to $x\le k$, state the largest value of $k$, in exact form, for which the function ${{\mathrm{f}}^{-1}}$ exist.

Using the domain from part (ii), define ${{\mathrm{f}}^{-1}}$ in a similar form.

Using the domain from part **(ii)**, define ${{\mathrm{f}}^{-1}}$ in a similar form.

Solve ${{\mathrm{f}}^{-1}}(x)=\mathrm{f}(x)$.

In the rest of the question, the domain of f is as originally defined.

The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto -{{x}^{3}},\mathrm{ }x\in \mathbb{R},\mathrm{ }x>0$

Find an expression for $\mathrm{fg}\left( x \right)$.

Modulus Functions

Functions

H2 Math

International Baccalaureate Higher Level Mathematics

2016 PJC P2 Q3

2016 PJC P2 Q3

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