How to find the domain of a function : a brief introduction to the theory of elementary functions
Undoubtedly, an essential attribute of any function is its domain. This article contains the most necessary initial information that answers the questions: ‘what is the domain of a function?’ and ‘how to find the domain of a function ?’ From it, you will learn what it is, how it is designated and what are the domains of the basic elementary functions (constant, root, exponential functions, etc.). This knowledge is quite useful when we are faced with the assignment aimed to determination of the domain of the function of different more complicated forms.
A numeric function with domain D is the equation, in which each number ‘x’ in the set D is compared with some number ‘y’, which depends on ‘x’, according to the specific rule of each concrete function. The domain of the function is the set of all values of the argument, which is given by the function. In other words, all values that can take the independent variable (argument) form the domain of the function. The domain of the function is also called the admissible domain of the function. However, one has to use official terminology in her term paper with an eye to eschew misconceptions with pedagogues and mistakes that can obscure one’s train of mathematical thought. The area of the function is usually denoted D(y). The dependence of the variable ‘y’ is a function of the variable ‘x’ if every value of ‘x’ corresponds a single value of ‘y’. The variable ‘x’ is called the independent variable or argument, and the variable ‘y’ – dependent variable. The equation: y = f(x) describes the function. The value of the function is the value of ‘y’, corresponding to a given value of ‘x’. In order to identify the domain of the function f, it is usually used the brief notation of the form D (f). Additionally, it should be noted that some specific functions, such as the trigonometric functions and inverse trigonometric functions, could be recorded in other forms. Therefore, for example, one can find a record D (sin), which represents the domain of the sine function. Of course, it can be rewritten as the D (f), where f is the sine function.
Therefore, if the domain of the function is the set of X, then the accepted entry form is D (f) = X. For example, the domain of arcsine (inverse sine function is referred to as arcsine) has a numerical interval [-1, 1], therefore, it can be written as D (arcsine) = [- 1, 1].
How to find the domain of a function : basic elementary functions
From the definition of the function, it is clear that the domain of the function is an integral part of the function, which is given along with the function. Thereby, the domain of the function is usually indicated initially. For example, at algebra lessons, students sequentially study properties of various functions: direct proportionality, linear function, the function y = x2, etc., and usually the domains of these functions are referred to as mathematical properties of these functions. Therefore, let us compose a succinct list of the elementary functions, which is composed according to the principles of the modern case study analysistogether with a brief explanation of the basic properties of each specific function:
- How to find the domain of a function : the constant function. The constant function, as is known, is given by the equation: y = C (or f (x) = C), where C is a real number. It assigns to each real value of the argument the specific value of the function that is equal to C. Thus, the domain of the constant function is the set of all real numbers R. For example, the domain of the constant function y = -7 (or f (x) = – 7) is the set of all real numbers (D (f) = (- ∞, + ∞) or D (f) = R.
- How to find the domain of a function : the power function. The power function is given by y = xa, therefore, f (x) = xa, where ‘x’ is a variable extent in the base and ‘a’ is a number in the exponent. The domain of the power function depends on the value of the exponent. If ‘a’ is a positive integer, then the domain of the function is the set of real numbers, which is the same as (-∞, + ∞). For non-integer real positive exponents D (f) = [0, + ∞). If ‘a’ is a negative integer, the domain of the function is a set (-∞, 0) (0, + ∞). For all other real negative numbers, the domain of the power function is a numerical interval (0, + ∞). For a = 0, the power function ‘y = x’ is defined for all real values of ‘x’, except x = 0, due to the fact that we have not defined 00. A number of non-zero at zero degrees, as is known, is equal to unity. Therefore, if a = 0, we have a function y = x0 = 1 on the domain of the power function (-∞, 0) (0, + ∞).
- How to find the domain of a function : y = ax. This function is given by the specific exponential function y = ax, where the variable ‘x’ is the exponent and the base is the number ‘a’, which is greater than zero and not equal to unity. It is determined on the set of real numbers. This statement means that the domain of the exponential function is a set of R. For example, let us examine the exponential functions: y = (1/8)x, y = ex, y = 1745x, y = (251/2)x. For all these functions the domain of the function ranges from -∞ to + ∞.
- How to find the domain of a function : the logarithmic function. The logarithmic function is the function with base ‘a’, where a> 0 and a ≠ 1. It is defined by the formula y = logax on the set of positive real numbers. The logarithm to the base ‘a’ is designated as a loga. The logarithm to the base ‘e’ is defined as ln, whereas the logarithm to the base ‘10’ is defined as a lg. The domain of the logarithmic function (or as they say, the domain of the logarithm) is the set of all positive real numbers. Therefore, D (loga) = (0, + ∞), in particular, D (ln) = (0, + ∞) and D (lg) = (0, + ∞). In addition, it should be noted that the domain of the logarithmic function is often called ‘the domain of the logarithm’. Thus, one should not be confused if she meets this term in specialized mathematical literary sources. In fact, the knowledge of this fact is quite useful in accomplishing various student’s assignments, such as formatting an essay about the logarithmic functions. Here are some examples. Let us examine the following logarithmic functions: y = log16x, y = log17x, y = ln x. The domain of all these functions is the set (0, + ∞).
- How to find the domain of a function : the trigonometric functions. The function, which is given by the formula y = sinx, is called sine. Obviously, it is denoted by sin and is defined on the set of all real numbers. Thus, the domain of the sine is the set of all real numbers, or, in other words, D (sin) = R. Similarly, the function defined by the formula y = cosx, is called cosine. It is denoted by cos and it is defined on the set R. The domain of cosine function is the set of all real numbers: D (cos) = R. The functions defined by the formulas y = tg x and y = ctg x, are called tangent and cotangent respectively. They are designated as tg and ctg. The domain of the tangent is the set of all real numbers except the numbers: (π/2) + πk, k belongs to the set Z. Therefore, the domain of the cotangent is the set of all real numbers except the numbers: πk, k belongs to the set Z. Thus, if ‘x’ is an argument of functions tangent and cotangent, the domains of the tangent and cotangent include all numbers ‘x’: x belongs to the set R, x ≠ (π/2) + πk; k belongs to the set Z and x belongs to the set R, x ≠ πk, k belongs to the set Z.
- How to find the domain of a function : the square root function. The domain of the square root function depends on the parity index. If n is an even number (n = 2m), then the domain of the square root function is the set of all non-negative real numbers. If the index of the root is an odd number greater than 1 (n = 2m + 1), then the domain of the square root function is the set of all real numbers. Therefore, the domain of each square root function: y = x1/2, y = x1/6 is the set of numbers [0, + ∞). The domain of each square root function: y = x1/3, y = x1/7 is the set of numbers (−∞, +∞).
- How to find the domain of a function : the inverse trigonometric functions. The function, which is given by the formula y = arcsin x and considered on the interval [-1, 1], is called the inverse sine. It is denoted as ‘arcsin’. From this definition it is obvious that the domain of the inverse sine is the set [-1, 1] or, in other words, D (arcsin) = [- 1, 1]. Similarly, the function that is given by the formula y = arcos x and considered on the interval [-1, 1] is called the inverse cosine function. It is designated as ‘arccos’. Thus, the domain of the inverse cosine function is the interval [-1, 1]. D (arccos) = [- 1, 1]. Functions that are given by the form y = arctg x and y = arcctg x and considered on the set of all real numbers, are called the arctangent and inverse cotangent respectively, and are indicated by ‘arctg’ and ‘arcctg’. The domain of the arctangent and arccotangent is the whole set of real numbers R. Therefore, D (arctg) = R and D (arcctg) = R.
The history of the concept through the centuries
Firstly, we should note that in the mathematical treatises written by first scholars who have examined this problem such as, the famous physicist Daniel Bernoulli and mathematician Gottfried Leibniz, the conception of function occurs only applied to specific practical examples. In fact, general formulations of the conception of the function occurred for the first time only in the XVIII century. The next important stage in the evolution of the mathematical thought is connected with the famous Swiss scholar – Leonhard Euler. Doubtlessly, he is one of the most illustrious scholars in the history of mathematics. Of course, each literary essay about the history of this algebraic conception contains mentions about this mathematical genius and his well-known mathematical treatises. Therefore, Leonhard Euler in 1750 proposed two different definitions of the term ‘function’. In his famous Introduction, he postulates that a function is every analytic expression containing ‘x’ or in other words, it is any expression consisting of the logarithms, trigonometric functions, etc. However, a few years later he proposed the common division of functions for algebraic and transcendental functions. Simultaneously, in his works we can find a couple of examples in which the function y(x) is defined by the fact that in the rectangular coordinate system the function is defined as an elementary curve that is drawn simply from the hand (it is a so-called ‘libera manu ducta’). Lagrange in his Theorie des fonctions analytiques, which was published in 1800, severely restricts the concept of function, reducing it to the so-called analytic functions defined by a series with respect to the value of ‘x’. According to his postulates, the smallest segment of an analytic function in the sense of Lagrange’s definition, completely defines it in its entirety. In fact, this property is in complete contradiction with the properties of functions in the sense of the second definition of the Leonhard Euler: every segment of the function can be extended arbitrarily. Without a doubt, one has to reflect this significant peculiarity in her grant proposal if it contains facts about the history of this conception. Therefore, we can state that the works of these mathematicians answered virtually all fundamental questions of elementary algebra, such as ‘what is the function?’, ‘what is the graphical pattern of the specific function?’, ‘how to find the domain of a function ?’, etc. The considerable development of the general theory of complex functions, which became gradually for about the next three decades, the common heritage of all mathematicians, had started around the 1830s. In fact, Augustin Louis Cauchy, Karl Theodor Wilhelm Weierstrass, and Georg Friedrich Bernhard Riemann made the most significant contribution to the evolution of this branch of mathematical science. Thus, the history of development of this idea was finished and mathematicians moved on to the study of more complicated mathematical concepts.
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