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Functions

A function is a relationship between two sets of numbers. We may think of this as a mapping; a function maps a number in one set to a number in another set. Notice that a function maps values to one and only one value. Two values in one set could map to one value, but one value must never map to two values: that would be a relation, not a function.

For example, if we write (define) a function as:

f(x) = x2

then we say:

'f of x equals x squared'

and we have

f( - 1) = 1

f(1) = 1

f(7) = 49

f(1 / 2) = 1 / 4

f(4) = 16

and so on.

Exponentiation

Exponentiation is a mathematical operation, written as aⁿ, involving two numbers, the base 'a' and the exponent 'n'.

When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a:

Just as multiplication by a positive integer corresponds to repeated addition:

The exponent is usually shown as a superscript to the right of the base.

The power aⁿ can be defined also when n is a negative integer, for nonzero a. No natural extension to all real a and n exists, but when the base a is a positive real number, aⁿ can be defined for all real and even complex exponents n via the exponential function e^z.

Trigonometric functions can be expressed in terms of complex exponentiation.

Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations.

Graphs of y=ax for various bases a: base 10 (green), base e (red), base 2 (blue), and base ½ (cyan). Each curve passes through the point (0,1) because any nonzero number raised to the power 0 is 1. At x=1, the y-value equals the base because any number raised to the power 1 is itself.

Algebra

Algebra is the branch of mathematics that concerns with the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, and number theory, algebra is one of the main branches of pure mathematics.

Elementary algebra introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. Algebra is much broader than elementary algebra. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields, studied in the area of mathematics called abstract algebra.

In 1545, the Italian mathematician Girolamo Cardano published Ars magna -The great art, a 40-chapter masterpiece in which he gave for the first time a method for solving the general quadratic equation.