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

(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)

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

      (2.13)

      that complements Eq. (2.9).

Another essential function is the (natural) exponential, e^x – 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.

      (2.14)

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

(2.15)

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

      (2.16)

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/e^y). A generalization of Eq. (2.15) reads

      (2.17)

where x₁ =