Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \left| x+20 \right|$, $x

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \left| x+20 \right|$, $x\in \mathbb{R}$, $\mathrm{g}:x\mapsto x+\frac{\alpha }{x}$, $x\in \mathbb{R}$, $x>0$. Explain why the composite function $\mathrm{gf}$ does not exist. [2] Determine the range of values of $\alpha $ such that ${{\mathrm{g}}^{-1}}$ exists. [2] For the rest of the question, it is given that $\alpha =4$ and the domain of $\mathrm{g}$ is further restricted to $x\ge \beta $. Determine the least value of $\beta $ for which ${{\mathrm{g}}^{-1}}$ exists; using this value of $\beta $, hence find ${{\mathrm{g}}^{-1}}\left( x \right)$ and state its domain. [4]

Accepted Answer

(a) gf does not exist; (b) α < 0; (c) β = 2, g⁻¹(x) = 2/√(1-x), Dg⁻¹ = [4, ∞)

HCI Functions Tutorial Q9

HCI Functions Tutorial Q9

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