<ol class="abc-list"><li><div class="list-item-paragraph">Given the functions</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{f}:x\mapsto 3{{x}^{3}}-4{{x}^{2}}-1$, $x\ge 1$,</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{g}:x\mapsto {{\mathrm{e}}^{2-0.25{{x}^{2}}}}$, $x\ge -3$.</div><ol class="i-list"><li><div class="list-item-paragraph">Explain why the functions ${{\mathrm{f}}^{-1}}$ and $\mathrm{gf}$ exist,</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the exact value of ${{\mathrm{f}}^{-1}}\left( -1 \right)$,</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the range of $\mathrm{gf}$, giving your answer in exact form.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">The functions $\mathrm{f}$ and $\mathrm{h}$ are defined by</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{f}:x\mapsto {{x}^{2}}-{{\lambda }^{2}}$, $x\le -\lambda $,</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{h}:x\mapsto \sqrt{x+{{\lambda }^{2}}}$, $x\ge 0$.</div><div class="list-item-paragraph">where $\lambda >0$. Define the following functions</div><ol class="i-list"><li><div class="list-item-paragraph">${{\mathrm{f}}^{-1}}$.</div></li><li><div class="list-item-paragraph">$\mathrm{hf}$.</div><p style="text-align: right;">[5]</p></li></ol></li></ol>

$\mathrm{f}:x\mapsto 3{{x}^{3}}-4{{x}^{2}}-1$, $x\ge 1$,

$\mathrm{g}:x\mapsto {{\mathrm{e}}^{2-0.25{{x}^{2}}}}$, $x\ge -3$.

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

$\mathrm{f}:x\mapsto {{x}^{2}}-{{\lambda }^{2}}$, $x\le -\lambda $,

$\mathrm{h}:x\mapsto \sqrt{x+{{\lambda }^{2}}}$, $x\ge 0$.

where $\lambda >0$. Define the following functions

Modulus Functions

Functions

H2 Math

International Baccalaureate Higher Level Mathematics

2010 ACJC CT2

2010 ACJC CT2

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