The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{e}^{x}}-7x,x\lambda $, where&

Question

The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{e}^{x}}-7x,x>\lambda $, where $\lambda $ is a real constant. Find the exact minimum value of $\lambda $ such that the inverse of $\mathrm{g}$ exists.[2] Using the value of $\lambda $ found in (i), find ${{\mathrm{g}}^{-1}}\left( 1 \right)$,[2] sketch the graphs of $y=\mathrm{g}\left( x \right)$, $y={{\mathrm{g}}^{-1}}\left( x \right)$ and $y=\mathrm{g}{{\mathrm{g}}^{-1}}\left( x \right)$ on a single diagram.[3]

Accepted Answer

(i) $\lambda = \ln 7$, (ii) $\mathrm{g}^{-1}(1) = 3.13$

2015 VJC P1 Q4

2015 VJC P1 Q4

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