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The R Book by Michael J. Crawley

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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 xb the explanatory variable is raised to the power b. In ex the explanatory variable appears as a power – in this special case, of e = 2.71828, of which x is the exponent. The inverse of ex 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: x0 = 1.
  • One raised to any power is still 1: 1x = 1.
  • Infinity plus 1 is infinity: ∞ + 1 = ∞.
  • One over infinity (the reciprocal of infinity, ∞−1) is zero: images = 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: xb = images.
  • Fractional powers are roots: x1/3 = .
  • The base ...

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