Prefixes and suffixes are both kinds of affixes. That is, they are word parts that attach to the beginning or end of a word or word base (a word stripped down to its simplest form) to produce a related word or an inflectional form of a word. Examples are in- in informal and both re- and -ing in reporting.
A third kind of affix is called an infix. Infixes are inserted into a word or word base. English uses very few infixes, but a couple examples are the plural-making s in words like cupsful and passersby, and various swear words, like damn in informal constructions like guaran-damn-tee.
A combining form is a form of a word that only appears as part of another word. There are a number of kinds of combining forms, each classified by what kind of word results when the form is used. For example, -wise in clockwise is an adverb combining form; -like in birdlike is an adjective combining form; -graph in photograph is a noun combining form; and -lyze in electrolyze is a verb combining form.
Combining forms are similar to affixes but can have a bit more lexical substance to them. Unlike affixes, combining forms are substantial enough to form a word simply by connecting to an affix, such as when the combining form cephal- joins with the suffix -ic to form cephalic. A combining form can also differ from an affix in its being derived from an independent word. For example, para- is a combining form in the word paratrooper because in that word it represents the word parachute. Para- is a prefix, however, in the words paranormal and paramedic. A combining form can also be distinguished historically from an affix by the fact that it is borrowed from another language in which it is descriptively a word or a combining form, such as the French mal giving English the mal- in malfunction.
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