## Mathematical Functions

For the kinds of functions you will meet in statistical computing there are only three mathematical rules that you need to learn: these are concerned with powers, exponents and logarithms. In the expression *x*^{b} the explanatory variable is raised to the **power** *b*. In *e*^{x} the explanatory variable appears as a power – in this special case, of e = 2.71828, of which *x* is the **exponent**. The inverse of *e*^{x} is the **logarithm** of *x*, denoted by log(*x*) – note that all our logs are to the base e and that, for us, writing log(*x*) is the same as ln(*x*).

It is also useful to remember a handful of mathematical facts that are useful for working out **behaviour at the limits**. We would like to know what happens to *y* when *x* gets very large (e.g. *x* → ∞) and what happens to *y* when *x* goes to 0 (i.e. what the intercept is, if there is one). These are the most important rules:

- Anything to the power zero is 1:
*x*^{0} = 1.
- One raised to any power is still 1: 1
^{x} = 1.
- Infinity plus 1 is infinity: ∞ + 1 = ∞.
- One over infinity (the reciprocal of infinity, ∞
^{−1}) is zero: = 0.
- A number bigger than 1 raised to the power infinity is infinity: 1.2
^{∞} = ∞.
- A fraction (e.g. 0.99) raised to the power infinity is zero: 0.99
^{∞} = 0.
- Negative powers are reciprocals:
*x*^{−b} = .
- Fractional powers are roots:
*x*^{1/3} = .
- The base ...