The function p(x) = (cos(2πx), sin(2πx)) maps the half-open interval [0, 1) in R to the unit circle in R². Due to its periodic nature, the function is not one-to-one, thus lacking an inverse outside the specified interval. The range of p(x) is defined as {(x,y) | x² + y² = 1}, which serves as the domain for the inverse function. The inverse function is given by p⁻¹(x,y) = arccos(x)/(2π) for y ≥ 0 and 1 - arccos(x)/(2π) for y < 0.
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