The artist's work can fall into any of the categories you mentioned, depending on the intent and the nature of the piece. For example:
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Only Utilitarian: A piece designed specifically for a practical purpose, like a kitchen tool or furniture.
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Only Aesthetic: A work created primarily for visual enjoyment, such as an abstract painting that serves no functional purpose.
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Both Utilitarian and Aesthetic: A piece like a beautifully designed chair that is not only functional but also serves as a decorative object.
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Neither Utilitarian nor Aesthetic: An artwork that may not have a practical use and may also not evoke any visual pleasure, such as a conceptual piece meant to provoke thought rather than admiration.
To determine which category applies to a specific piece of art by an artist, you'd need more context about the piece in question.