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\bf{Question}

\quad Which of the following functions $\phi$ are
reparametrizations from the open interval \break $J=(-1,1)$ to the
open interval $I=\phi(J)\,$? $$\phi(t)=\hspace{1cm}{\rm (i)}\
t+2t^4\hspace{1cm}{\rm (ii)}\ t+t^5\hspace{1cm} {\rm (iii)}\
t^3+\sin^3\pi t.$$



\bf{Answer}

\begin{description}
\item{(i)}
NO: not injective (e.g. $\varphi(0) =0$ and $\varphi \left (
-\frac{1}{\sqrt[3]{2}} \right ) =0$, $-1 < -\frac{1}{\sqrt[3]{2}}
< 0 <1$.

\begin{center}
$J=(-1,1)$

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\item{(ii)}
YES: $\varphi'(t) = 1+5t^4 >0, \ \forall t$

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\item{(iii)}
NO: $\varphi'(0) =0$ so if $\psi = \varphi^{-1}$ then $\psi$ could
not be differentiable at $0=\varphi(0)$.
\end{description}


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