\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(f : \,]0, +\infty[ \to \mathbb{R}\) defined by \(f(x) = x \ln x - x\). Determine the image \(f(\,]0, +\infty[\,)\) and identify on which sub-interval \(f\) is bijective onto its image.
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