The function $\mathrm{f}$ and $\mathrm{g}$ are defined as follows: \(\mathrm{f}:x\mapsto {{\left

Question

The function $\mathrm{f}$ and $\mathrm{g}$ are defined as follows: $\mathrm{f}:x\mapsto {{\left( 2x-1 \right)}^{2}}-2$, $ \qquad x<-1$ $\mathrm{g}:x\mapsto \ln \left( x+a \right)$, $x>-1$ Define ${{\mathrm{f}}^{-1}}$ in a similar form.[3] State the value of $a$ such that the range of $\mathrm{g}$ is $\left( 0,\infty \right)$.[1] Show that the composite function $\mathrm{gf}$ exists, and find the range of $\mathrm{gf}$, giving your answer in terms of $a$.[2]

Accepted Answer

(a) ${{\mathrm{f}}^{-1}}:x\mapsto \frac{1}{2}-\frac{1}{2}\sqrt{x+2},x>7$ (b) $a=2$ (c) ${{R}_{\mathrm{fg}}}=\left( \ln \left( a+7 \right),\infty \right)$

2010 HCI P1 Q3

2010 HCI P1 Q3

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