Algebraic Functions
Algebraic functions are mathematical functions that can be constructed using a finite number of algebraic operations, such as addition, subtraction, multiplication, division, and taking roots. These functions can be expressed as a ratio of two polynomials and can include constants and variables. They are fundamental in algebra and are used to model various real-world phenomena.
5 Key excerpts on "Algebraic Functions"
- Craig Adam(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...2 Functions, formulae and equations Introduction: Understanding and using functions, formulae and equations The ability to use formulae correctly and to manipulate them effectively is a fundamental mathematical skill of the forensic scientist. The material in this chapter forms the basis of much that follows in later chapters as it includes the essential algebraic skills used in many forensic applications. You will find that the rules for mathematical manipulation and formula substitution are best learned through practice and will become embedded in your mind by applying them to many different examples and symbolic notations. You will also benefit from trying to keep your mathematical work clear and accurate as you write it down, as this will minimize the chances of errors through mistranscription of detail. Further, it is beneficial to develop some understanding of the vocabulary of mathematics, as this will help you learn new material as you progress through this book. A function is a mathematical operation or set of operations that maps one set of numbers on to another; so, for example y = 2 x + 1 maps the numbers represented by x on to the set obtained by multiplying by 2 then adding 1. Since the function maps x on to y we usually write this as y(x) = 2 x + 1; the (x) bit has no relevance to working out or manipulating the function in practice! There are many functions we will meet in this and the two subsequent chapters that describe more complicated relationships such as the exponential function y(x) = e x and the trigonometric functions e.g. y(x) = sin x. However, the basic interpretation and mathematical processing of all functions remains the same. When functions represent real behaviour or properties and are used with measurable quantities or to interpret experimental data, we tend to call them formulae. The quantities in a formula usually have units and relate to real situations, e.g...
eBook - ePub- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 3 Algebraic Operations CHAPTER 3 ALGEBRAIC OPERATIONS An important part of understanding any topic is knowing the vocabulary, and this is especially true in mathematics. This chapter introduces the vocabulary of algebra. OPERATIONS WITH ALGEBRAIC EXPRESSIONS Algebra uses letters to represent numbers. For example, in the formula for the surface area of a sphere, A = 4π r 2, A stands for the surface area, r stands for the radius of the sphere, and π is the Greek letter pi, with a value of approximately 3.14. The letters A and r can represent many numbers, so they are called variables. The number 4 and letter π each represents only one number, so they are called constants. Terms are constants, variables, or a product or quotient of constants and variables. For example, 5, 4 x, –2 xy, and are all terms. The constant part of the term is called a coefficient, so the coefficient of 4 x is 4. Algebraic expressions consist of one or more terms. They can contain mathematical symbols such as +, (), or. Examples of algebraic expressions include Typically, we substitute numbers for the variables in an expression and then do the arithmetic to get a numerical answer. This is called evaluating the expression. EXAMPLE Evaluate 2 x – y for and y = –5. SOLUTION. EXAMPLE Evaluate |4 a 2 + b – 1| for a = 0 and. SOLUTION. Equivalent expressions always have the same value for all replacements of variables. The expression 2 x – 3 is equivalent to 2 x + (–3), no matter what value x has. Likewise,. Try it. Several properties, discussed next, are helpful in determining whether expressions are equivalent. Commutative Properties The commutative property allows us to change the order when adding or multiplying. Addition: For any real numbers a and b, a + b = b + a. Thus, we can say 3 + 5 = 5 + 3. Multiplication: For any real numbers a and b, ab = ba. Thus, 2 · 7 = 7 · 2, which can also be written as 2(7) = 7(2)...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...Division of polynomials is important since it is related to finding factors of polynomials. If 6 is divided by 2, the remainder is 0, so 2 is a factor of 6. Similarly, you can tell whether a polynomial is a factor of another polynomial by determining whether their division produces a remainder of 0. Solving a polynomial equation p (x) = 0 often depends on finding factors of p (x). POLYNOMIAL FUNCTIONS AND RELATED TERMS The function is not a polynomial function since it contains. The function is not a polynomial function since it contains. Since each of the exponents of x in is a whole number, p (x) is a polynomial function. DEFINITION OF POLYNOMIAL FUNCTION A polynomial function, p, is a function that can be written in the form where n is a positive integer called the degree of the polynomial, provided that a n ≠ 0. Each member of the sum is a term of the polynomial. The number constants a n, a n – 1, a n – 2, …, a 1, a 0 are coefficients. The leading coefficient is a n, and the constant term is a 0. Note: Polynomial functions should be written in descending order, starting with the term of the highest power. Any term missing in the polynomial means that 0 is the coefficient of that term. If • The degree is 4. • The leading coefficient is 7, and the constant term is 9. • The values of the coefficients are, and a 0 = 9. NAMING THE PARTS OF A DIVISION EXAMPLE In the division example 8 ÷ 5, 8 is the dividend, 5 is the divisor, 1 is the quotient, and 3 is the remainder. In general, DIVISION OF POLYNOMIAL FUNCTIONS The division relationship for numbers is true also for polynomials...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...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, x 1, x 2, …, x n, then a multivariate function arises, y ≡ y { x 1, x 2, …, x n, }. 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 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 n denote (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...
eBook - ePubMATLAB® Essentials
A First Course for Engineers and Scientists
- William Bober(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...6 Roots of Algebraic and Transcendental Equations 6.1 Introduction In the analysis of various engineering problems, we are often faced with a need to find roots of equations whose solution is not easily found analytically. Given a function f (x), the roots of the function are the values of x that makes f (x) = 0. For example, the equation f (x) = a x 2 + b x + c = 0 (6.1) where a, b, and c are constants, is an equation that we are all familiar with. The values of x that satisfy the equation are the roots of f (x). We even have a formula for the roots, which are x = − b ± b 2 − 4 a c 2 a (6.2) We see that there are two roots, x 1 and x 2, where x 1 = − b + b 2 − 4 a c 2 a, x 2 = − b − b 2 − 4 a c 2 a (6.3) More complicated examples include n th degree polynomials and transcendental equations containing trigonometric, exponential, or logarithm functions. In this chapter, we discuss the search method for obtaining a small interval in which a root lies. We then discuss MATLAB ® ’s fzero and roots functions, which may be used to obtain a more accurate value for the roots of type of equations just stated. 6.2 Search Method In the search method, we seek a small interval that contains a real root. This only gives an approximate value for the real root. Once an interval in which a real root lies has been established, several different methods, including the Bisection method, Newton–Raphson method, and MATLAB’s fzero and roots functions, can be used to obtain a more accurate value for the real root. In this book, we will give a brief discussion of the Bisection method, but emphasize MATLAB’s fzero and roots functions. The search method is especially useful if there is more than one real root...
