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Question 3.35: Prove that the mapping f: R → R given by f(x)={x^4  if x ≥ 0......

Prove that the mapping  f:\mathbb{R}\to\mathbb{R}  given by

f(x)={\left\{\begin{array}{l l}{x^{4}}\quad{{\mathrm{if}}\quad x\geq 0;}\\ {x(2-x)\quad{\mathrm{if}}\quad x\lt 0,}\end{array}\right.}

is a bijection, and find its inverse.

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It suffices to find  g:\mathbb{R}\rightarrow\mathbb{R}  such that   f\circ g=g\circ f=\operatorname{id}_{\mathbb{R}}.  Then f  is a bijection with  \,f^{-1}=g.  Now if  y=x^{4}  we have  \textstyle{{x}}={\mathcal{y}}^{1/4}  so we can define

g(x)=x^{1/4}\quad\mathrm{~if~}\quad x\geqslant0.

Also, if y = x(2 – x) then x² – 2x +y = 0 and we  have  x=1\pm\sqrt{(}1-y).

Since  1+\sqrt(1-y)\gt 0  we must choose  x=1-\sqrt(1-y)  and define

g(x)=1-\sqrt{\left(1-x\right)}\quad\mathrm{if}\quad x\lt 0.

Then it is easy to check that f [g(x)] = x and g[f(x)] = x for every  x\in\mathbb{R},   so that g is the inverse of f.

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