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What do we call functions that take an input and make it larger, so that $f(x) > x$ for every $x$ in its domain?

A simple example would be $f(x) = x + 1$ for $x \in \mathbb R$ but there are other well-known examples such as $f(x) = \sqrt x$ for $x \in (0,1)$.

Note that such functions need not be increasing; we can even construct a decreasing function such as $f(x) = 1000 - x$ on the domain $x \in (0,1)$ which clearly satisfies $f(x) > x$.

This question is somewhat related to the idea of prefixpoints and postfixpoints but that doesn't quite capture it; e.g. a postfixpoint satisfies $x \leq f(x)$ whereas I am demanding something stronger, that $x < f(x)$ for every point in the domain.

Similarly, what is the name for functions for which $f(x) < x$?

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    I don't think there is a name, nor do a see a reason to have one. You might say $f$ is strictly greater than the identity function. – 2017-02-26
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    @ThomasAndrews Take the following, informal statement: *it's a common misconception that square rooting a number always makes it the same or smaller - but if you square root a number between 0 and 1, it actually gets larger!* It would surprise me if that couldn't be expressed more formally... I also thought the terminology would have come in useful when discussing e.g. contraction mappings (to the left of an attractive fixed point, the function is *makebiggier*, to the right it is *makesmallier*). But it seems people get by without it! – 2017-02-26
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    "If you square root a number..." is also, of course, not normal math-speak. We'd normally say, "If you take the square of a number...' – 2017-02-26

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