F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

Among the most useful quantitative relationships, polynomial functions stand up – of the form P n{ x } ≡ a n x n+ a n − 1 x n − 1+ ⋯ + a 1 x + a 0, where a 0, a 1, …, a n−1, and a ndenote (constant) real coefficients and n denotes an integer number; a rational function appears as the ratio of two such polynomials, P n{ x }/ Q m{ x }, where subscripts n and m denote polynomial degree of numerator and denominator, respectively. Any function y { x } satisfying P { x } y m + Q { x } y m−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,

(2.1)

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 Τ , i.e. f { x + nΤ } = f { x } with n integer, then such a function is termed periodic of period Τ ; a common example is sine and cosine with period 2 π rad , as well as tangent with period π 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 π 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

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

or else

(2.6)

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

(2.7)

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

(2.8)

in general terms, one concludes that

(2.9)

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

(2.10)

to readily obtain

(2.11)

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

(2.12)