Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \ln \left( 4{{x}^{2}}-1

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \ln \left( 4{{x}^{2}}-1 \right)$ for $x\in \mathbb{R}$, $x$\mathrm{g}:x\mapsto 2x-1$ for $x\in \mathbb{R}$, $x\ge 3$. Show that $\mathrm{f}$ has an inverse. [2] Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$. [3] Explain why the composite function ${{\mathrm{f}}^{-1}}\mathrm{g}$ exist. [2] Find the exact value of $x$ when ${{\mathrm{f}}^{-1}}\mathrm{g}\left( x \right)=-7$. [2]

Accepted Answer

(ii) ${{{\mathrm{D}}_{{{\mathrm{f}}^{-1}}}}={{\mathrm{R}}_{\mathrm{f}}}=\left[ 0,\infty \right)}$ (iii) Since ${{{\mathrm{R}}_{\mathrm{g}}}\subseteq {{\mathrm{D}}_{{{\mathrm{f}}^{\mathrm{-1}}}}}}$, thus ${{\mathrm{f}}^{-1}}\mathrm{g}$ exist. (iv) $x=\frac{1+\ln 195}{2}$

2020 RI J1 CT Q7

2020 RI J1 CT Q7

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