ASRJC Functions Tutorial Q9
The function $\text{f}$ is defined by $\text{f}:x\mapsto \left| 3+x-2{{x}^{2}} \right|$ for $x\in \mathbb{R}$, $x\le k$.
(i)
Find the greatest value of $k$ such that $\text{f}$ has an inverse.
[1]
(ii)
Using the value of $k$ in part (i), find the inverse function ${{\text{f}}^{-1}}$.
[3]
(iii)
Sketch both $\text{f}$ and ${{\text{f}}^{-1}}$ on the same diagram. Is there any solution for $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$?
[3]
Another function $\text{g}$ is given by $\text{g}:x\mapsto {{\text{e}}^{-x}}$, $x\in \mathbb{R}$.
(iv)
Show that ${{\text{f}}^{-1}}\text{g}$ exists and find the range of this composite function.
[3]
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