The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \ln \left( {{x}^{2}}-x-2 \right)$, $c F

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \ln \left( {{x}^{2}}-x-2 \right)$, $c Find the values of $c$ and $d$ such that the range of $\mathrm{f}$ consists of all real values which are less than $\mathrm{e}$. Find an expression for ${{\mathrm{f}}^{-1}}\left( x \right)$ for all the values of $c$ and $d$ found in (a). Write down an expression for $\mathrm{h}\left( x \right)$ such that $\mathrm{hf}\left( x \right)={{x}^{2}}-x-2$.

Accepted Answer

(a) $c = -3.67$, $d = -1$ (b) $f^{-1}: x \mapsto \frac{1}{2} - \sqrt{e^x + \frac{9}{4}}, x < e, x \in \mathbb{R}$ (c) $h(x) = e^x$

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