 
Summary: CALCULATION OF FUNCTIONS
Using hand calculations, a hand calculator, or a com
puter, what are the basic operations of which we are
capable? In essence, they are addition, subtraction,
multiplication, and division (and even this will usually
require a truncation of the quotient at some point).
In addition, we can make logical decisions, such as
deciding which of the following are true for two real
numbers a and b:
a > b, a = b, a < b
Furthermore, we can carry out only a finite number
of such operations. If we limit ourselves to just addi
tion, subtraction, and multiplication, then in evaluat
ing functions f(x) we are limited to the evaluation of
polynomials:
p(x) = a0 + a1x + · · · anxn
In this, n is the degree (provided an 6= 0) and {a0, ..., an}
are the coefficients of the polynomial. Later we will
discuss the efficient evaluation of polynomials; but for
now, we ask how we are to evaluate other functions
