Curve and admissible change of variable

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Homework Statement

If I have the two curves

$\phi (t) = ( \cos t , \sin t )$ with $t \in [0, 2\pi]$
$\psi(s) = ( \sin 2s , \cos 2s )$ with $s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ]$

My textbook says that they are equivalent because $\psi(s) = \phi \circ g^{-1}(s)$ where $g^{-1} : [\frac{\pi}{4} , \frac{5 \pi}{4} ] \rightarrow [0, 2\pi]$ and $g^{-1}(s)= \frac{5\pi}{2} -2s$ (so $g^{-1}(5\pi /4) = 0$ and $g^{-1}(\pi /4) = 2\pi$ ).

I found instead $g^{-1}(s)= 2s - \frac{\pi}{2}$ ( so $g^{-1}(\pi /4) =0$ and $g^{-1}(5 \pi /4) = 2\pi$ )

Now the problem is, there are two admissible changes of variable and if I choose one or the other the way I go on the curve is opposite (because my $(g^{-1})' > 0$ and my textbook $(g^{-1})' <0$ ).

Now I don't understand if a change of variable can change the way of the curve.
Any help?

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Did you make a plot of the two curves ?