I was about to register a domain name which has some profanity in it.
Would the profanity have any influence on SEO, in particular would it harm the SEO of the site?
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Yes, the context of the word affects how strongly people react against its usage. We can truly understand the context, but search engines cannot fully understand. Google and other search engines have sets of filters to prevent profanity.
One example of how they do this is Google's "SafeSearch" filter. Granted, this is disabled by default and is intended for filtering "sexually explicit content", but individual can enable this feature. If your web pages have sexually explicit words, images, etc., users with this filter enabled will not see your web pages in the search results. Thus, these websites will lose some traffic.