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2 Function Features
ОглавлениеIf 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 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 2π 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
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,
(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
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
(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, 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.
(2.14)
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
(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/ey). A generalization of Eq. (2.15) reads
(2.17)
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.
(2.19)
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
(2.21)
after taking exponentials of both sides, where Eq. (2.15) supports
(2.22)
– while the definition of inverse function, applied three times, allows one to get
(2.23)
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. (2.20), with isolation of ln {x/y} retrieving the above result; hence, a logarithm transforms a quotient into a difference.
The concept of logarithm extends to bases other than e, say,
a‐based exponentials may then be taken of both sides to get
– since a‐based exponential and logarithm are inverse functions of each other. If b‐based logarithms are taken of both sides, then Eq. (2.28) becomes
(2.29)
in agreement with Eq. (2.25) and after application to Eq. (2.28) – which may, in turn, be combined with Eq. (2.27) to generate
(2.30)
upon isolation of loga x, one gets
Equation (2.31) may be used to convert the logarithm of (any number) x from base b to base a – at the expense of knowledge of logb a; in particular, one finds that
when a = 10 and b = e, with ln 10 equal to 2.302 59.
The concept of base other than e may indeed be extended to the exponential function itself – and coincides with a plain power, using a as base and the target function as exponent; furthermore, one may state that
since b‐based exponential and logarithm are inverse functions of each other, and Eq. (2.25) applies; after considering an analogue of Eq. (2.18) for the power of a power, one may redo Eq. (2.33) as
In particular, Eq. (2.34) has it that
(2.35)
for a ≡ 10 and b ≡ e – thus making a tool available to convert a natural exponential to a decimal exponential; note ln 10 appearing again as conversion factor, in parallel to Eq. (2.32).
Complex functions do often exhibit simple linear behaviors near specific finite value(s), or when their independent variable grows unbounded toward either −∞ or ∞; such a driving line – originally described by Apollonius of Perga in the Greek Antiquity, is termed asymptote, and represents a straight line tangent to the germane curve at infinity. A vertical asymptote is accordingly defined by
(2.36)
or
(2.37)
typical examples of a are the zeros of the denominator (or poles) of rational functions, or the value(s) that turn nil the argument of a logarithmic function. Oblique asymptotes abide, in turn, to
– which will, in particular, be horizontal if b = 0; division of both sides by x transforms Eq. (2.38) to
(2.39)
because 0/x = 0 for x ≠ 0 (as is the case) – where a/x becoming, in turn, negligible when x → 0 permits simplification to
If the limit described by Eq. (2.40) does not exist, then there is no oblique asymptote in that direction; otherwise, one may proceed and compute a from Eq. (2.38) via
(2.41)
where b obviously abides to Eq. (2.40). Although the concept of asymptote may be extended to other polynomial forms (e.g. quadratic) using essentially the same rationale, their determination (and usefulness) is far less common and rather limited.
When in the presence of two (or more, say, n) real values, one may define the most likely value, or arithmetic mean (referred to via subscriptarm, with denoting mean) as
by the same token, one can define a geometric mean (referred to via subscriptgem) as
The harmonic mean (referred to via subscriptham) satisfies
or, after taking reciprocals of both sides,
the aforementioned three means are useful in a great many problems – depending on the underlying mathematical nature of the data, so their relative location deserves further exploitation (as done below).
Consider, for simplicity, only two values x1 and x2; after realizing that
(2.46)
for being a square – with validity assured irrespective of the relative magnitude of x1 and x2, one may apply Newton’s binomial (to be considered shortly) to write
Upon addition of 4x1 x2 to both sides, Eq. (2.47) becomes
(2.48)
– where Newton’s binomial may again be invoked to support condensation to
If square roots are taken of both sides, then Eq. (2.49) transforms to
(2.50)
on the common assumption that both x1 and x2 are positive – whereas division of both sides by 2 unfolds
(2.51)
based on Eqs. (2.42) and (2.43), one concludes that
– i.e. the arithmetic mean of two numbers never lies below their geometric mean (being equal only when x1 = x2).
On the other hand, inspection of Eq. (2.44) vis‐à‐vis with Eq. (2.42) indicates that is the arithmetic mean of 1/x1 and 1/x2 (in the case of n = 2); hence, application of the result conveyed by Eq. (2.52) indicates that
After taking reciprocals of the left‐ and right‐hand sides, Eq. (2.53) gives rise to
(2.54)
or, after recalling Eq. (2.43),
Therefore, the harmonic mean of two numbers never exceeds the geometric mean (again encompassing only positive values), and coincides therewith again when x1 = x2.
When n = 2, another mean can be defined as
– known as logarithmic mean; Eq. (2.56) is often rephrased to
after taking advantage of Eq. (2.26). This logarithmic mean lies below the arithmetic and the geometric means, i.e.
where Eq. (2.55) was meanwhile taken advantage of. To prove so, it is convenient to insert Eqs. (2.42), (2.43), (2.45), and (2.57) pertaining to x1 and x2 so as to get
(2.59)
from Eq. (2.58), where factoring of x1 ≥ x2 > 0 (as per working hypothesis) in all sides gives rise to
after dropping x1 > 0 from all sides, and further multiplying and dividing the last side by x2/x1, Eq. (2.60) turns to
– where z denotes an auxiliary variable satisfying
A graphical account of Eq. (2.60) is provided in Fig. 2.3. Inspection of the curves therein not only unfolds a clear and systematic positioning of the various means relative to each other – in general agreement with (so far, postulated) Eq. (2.58) – but also indicates a collective convergence to x1 as x2 approaches it (as expected).
Figure 2.3 Variation of arithmetic mean (arm), logarithmic mean (lom), geometric mean (gem), and harmonic mean (ham) of positive x1 and x2 < x1, normalized by x1, as a function of their ratio, x2/x1.
To provide a quantitative argument in support of the graphical trends above, one may expand the middle left‐ and middle right‐hand sides of Eq. (2.61) via Taylor’s series (see discussion later), around z = 1, according to
where 1/2 in the left‐hand side was meanwhile splitted as 1 − 1/2; note that such series in z are convergent because 0 < z < 1, as per Eq. (2.62). Straightforward simplification of Eq. (2.63) unfolds
(2.64)
where z − 1 may be further dropped off both numerator and denominator of the middle side to give
after simplifying notation in Eq. (2.65) to
(2.66)
with the aid of
(2.67)
one may add ζ/2 to all sides to get
– where all terms in the denominator of the middle side are positive, whereas all terms (besides 1) in the right‐hand side are negative. Long (polynomial) division of 1 by 1 + ζ/2 + ζ2/3 + ζ3/4 + ⋯ (according to an algorithm to be presented below) allows further transformation of Eq. (2.68) to
(2.69)
where condensation of terms alike in the middle side unfolds
after having dropped unity from all sides, and then taken their negatives, Eq. (2.70) becomes
– which is a universal condition, since 1/12 < 1/8, 1/24 < 1/16, and so on in terms of pairwise comparison. Similar trends for the relative magnitude of the coefficients of similar powers would be found if the series were truncated after higher order terms – so one concludes on the general validity of Eq. (2.58), based on Eq. (2.71) complemented by Eq. (2.55).
