2
Function Features

If a relationship between two real variables, y and x, is such that y becomes determined whenever x is given, then y is said to be a univariate (real-valued) function of (real-variable) x; this is usually denoted as y?=?y{x}, where x is termed independent variable and y is termed dependent variable. The same value of y may be obtained for more than one value of x, but no more than one value of y is allowed for each value of x. If more than one independent variable exist, say, x1, x2, ., xn, then a multivariate function arises, y?=?y{x1,?x2,?.,?xn ,?}. The range of values of x for which y is defined constitutes its interval of definition, and a function may be represented either by an (explicit or implicit) analytical expression relating y to x (preferred), or instead by its plot on a plane (useful and comprehensive, except when x grows unbounded) - whereas selected values of said function may, for convenience, be listed in tabular form.

Among the most useful quantitative relationships, polynomial functions stand up - of the form Pn {x}?=?an xn ?+?an?-?1 xn?-?1?+???+?a1 x?+?a0, where a0, a1, ., an-1, and an denote (constant) real coefficients and n denotes an integer number; a rational function appears as the ratio of two such polynomials, Pn {x}/Qm {x}, where subscripts n and m denote polynomial degree of numerator and denominator, respectively. Any function y{x} satisfying P{x}ym +?Q{x}ym-1 +?? +?U{x}y?+?V{x} =?0, with m denoting an integer, is said to be algebraic; functions that cannot be defined in terms of a finite number of said polynomials, say, P{x}, Q{x}, ., U{x}, V{x}, are termed transcendental - as is the case of exponential and logarithmic functions, as well as trigonometric functions.

A function f is said to be even when f{-x} =?f{x} and odd if f{-x} =?-f{x}; the vertical axis in a Cartesian system serves as axis of symmetry for the plot of the former, whereas the origin of coordinates serves as center of symmetry for the plot of the latter. Any function may be written as the sum of an even with an odd function; in fact,

upon splitting f{x} in half, adding and subtracting f{-x}/2, and algebraically rearranging afterward. Note that f{x} +?f{-x} remains unaltered when the sign of x is changed, while f{x}?- f{-x} reverses sign when x is replaced by -x; therefore, (f{x} +?f{-x})/2 is an even function, while (f{x}?- f{-x})/2 is an odd function.

When the value of a function repeats itself at regular intervals that are multiples of some ?, i.e. f{x?+?n?} =?f{x} with n integer, then such a function is termed periodic of period ?; a common example is sine and cosine with period 2p rad, as well as tangent with period p rad (as will be seen below).

A (monotonically) increasing function satisfies , whereas a function is called (monotonically) decreasing otherwise, i.e. when ; however, a function may change monotony along its defining range.

If y?=?f{x}, then an inverse function f-1{y} may in principle be defined such that f-1{f{x}} =?x - i.e. composition of a function with its inverse retrieves the original argument of the former. The plot of f-1{y} develops around the x-axis in exactly the same way the plot of f{x} develops around the y-axis; in other words, the curve representing f{x} is to be rotated by p?rad around the bisector straight line so as to produce the curve describing f-1{y}.

Of the several functions worthy of mention for their practical relevance, one may start with absolute value, |x| - defined as

which turns a nonnegative value irrespective of the sign of its argument; its graph is provided in Fig. 2.1. It should be emphasized that |x| holds the same value for two distinct real numbers (differing only in sign), except in the case of zero.

Figure 2.1 Variation of absolute value, |x|, as a function of a real number, x.

It is easily proven that

based on the four possible combinations of signs of x and y, coupled with Eq. (2.2); by the same token,

after replacing y by its reciprocal in Eq. (2.3). On the other hand, the definition conveyed by Eq. (2.2) allows one to conclude that

or else

after taking negatives of both sides; based on the definition as per Eq. (2.2) and the corollary labeled as Eq. (2.5), one finds

upon replacement of x by x + y =?0. Equations (2.2) and (2.6) similarly support the conclusion

in general terms, one concludes that

after bringing Eqs. (2.7) and (2.8) together. On the other hand, one may depart from the definition of auxiliary variable z as

to readily obtain

after recalling Eq. (2.9), one may redo Eq. (2.11) to

with the aid of Eq. (2.10), where straightforward algebraic rearrangement unfolds

that complements Eq. (2.9).

Another essential function is the (natural) exponential, ex - i.e. a power where Neper's number (ca. 2.718 28) serves as basis; it is sketched in Fig. 2.2a. Note the exclusively positive values of this function - as well as its horizontal asymptote, viz.

The exponential function converts a sum into a product, i.e.

based on the rule of multiplication of powers with the same base; one also realizes that

pertaining to a difference as argument, and obtainable from Eq. (2.15) after replacement of y by -y (since e-y is, by definition, 1/ey). A generalization of Eq. (2.15) reads

where x1 = x2 = ? =?xn =?x readily implies

by virtue of the definition of multiplication as an iterated sum.

Figure 2.2 Variation of (natural) (a) exponential, ex, and (b) logarithm, ln x, as a function of a real number, x.

The inverse of the exponential is the logarithm of the same base, i.e. ln x for the case under scrutiny encompassing e as base; the corresponding plot is labeled as Fig. 2.2b. A vertical asymptote, viz.

is apparent (the concept of limit will be explored in due course); the plot of ln x may be produced from that of ex in Fig. 2.2a, via the rotational procedure referred to above. In terms of properties, one finds that

- so the logarithm converts a product to a sum; in fact, Eq. (2.20) is equivalent to

after taking exponentials of both sides, where Eq. (2.15) supports

- while the definition of inverse function, applied three times, allows one to get

as universal condition, thus guaranteeing validity of Eq. (2.20). If n factors xi are considered, then Eq. (2.20) becomes

should x1 = x2 = ? = xn = x hold, then Eq. (2.24) simplifies to

If y is replaced by 1/y in Eq. (2.20), then one eventually gets

- since ln?{x/y} + ln y = ln?{xy/y} = ln x as per Eq....