Off and on frequently appear as prefixes, but word structure can vary: Should the prefix be hyphenated to the root word, or should the entire word be a closed compound? This post lists examples of such terms.
Prefixed words, like compound words, go through an evolutionary process. Unlike as is the case with compounds, however, there is no open phase. Online, offbeat, and the like derive from the idea of being “on a line” or “off the beat,” for example, but they never existed as “on line” or “off beat.” (“On line” is a dialectal variant of “in line,” referring to standing in a queue, but this sense is distinct from the notion of an electronic link.) However, the former did start out in hyphenated form, transitioning during the 1990s as web browsing went mainstream (though a few publications and organizations remain holdouts), while offbeat was coined as a closed compound.
Closed off- and on- constructions are prevalent, but some hyphenated terms persist. Generally, however, if the word is a noun, it is closed. Consider the following: offshoot and offspring, and onlooker and onset. (However, off-ramp and on-ramp stubbornly remain hyphenated.)
Adjectives seem to be more of a mixed bag: Besides offbeat and online, closed compounds include offsetting and ongoing and the pairs offside and onside, offshore and onshore, and offstage and onstage (all of which are occasionally seen hyphenated). But note the pairs on-air and off-air and off-screen and on-screen, as well as off-color, off-key, off-limits, and off-white. (A few such terms, such as off-screen and on-screen, occasionally appear closed.)
Note that phrases beginning with off or on that serve to modify a noun are hyphenated before it, as in “off-the-cuff remarks” and “on-the-job injuries.” Treatment after the noun varies, however, according to whether the phrase is permanent or temporary. Off-the-cuff, which appears in dictionaries, is rendered as such after the noun (“remarks made off-the-cuff”), while “on the job” is not considered a standing phrase, so it is not hyphenated when it follows a noun (“injuries that occurred while an employee was on the job”).
How does one know the difference between such phrases? One keeps a list or consults a dictionary, or both. Unfortunately, one of these strategies, or a combination of the two, is essential also for confirming the style for terms prefixed by off or on.