Editing Function (mathematics) (section)

=== By a formula ===
Functions are often defined by an [[expression (mathematics)|expression]] that describes a combination of [[arithmetic operations]] and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain.
For example, in the above example, <math>f</math> can be defined by the formula <math>f(n) = n+1</math>, for <math>n\in\{1,2,3\}</math>.

When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the [[zero of a function|zeros]] of auxiliary functions. Similarly, if [[square root]]s occur in the definition of a function from <math>\mathbb{R}</math> to <math>\mathbb{R},</math> the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.

For example, <math>f(x)=\sqrt{1+x^2}</math> defines a function <math>f: \mathbb{R} \to \mathbb{R}</math> whose domain is <math>\mathbb{R},</math> because <math>1+x^2</math> is always positive if {{mvar|x}} is a real number. On the other hand, <math>f(x)=\sqrt{1-x^2}</math> defines a function from the reals to the reals whose domain is reduced to the interval {{closed-closed|−1, 1}}. (In old texts, such a domain was called the ''domain of definition'' of the function.)

Functions can be classified by the nature of formulas that define them:
* A [[quadratic function]] is a function that may be written <math>f(x) = ax^2+bx+c,</math> where {{math|''a'', ''b'', ''c''}} are [[constant (mathematics)|constants]].
* More generally, a [[polynomial function]] is a function that can be defined by a formula involving only additions, subtractions, multiplications, and [[exponentiation]] to nonnegative integer powers. For example, <math>f(x) = x^3-3x-1</math> and <math>f(x) = (x-1)(x^3+1) +2x^2 -1</math> are polynomial functions of <math>x</math>.
* A [[rational function]] is the same, with divisions also allowed, such as <math>f(x) = \frac{x-1}{x+1},</math> and <math>f(x) = \frac 1{x+1}+\frac 3x-\frac 2{x-1}.</math>
* An [[algebraic function]] is the same, with [[nth root|{{mvar|n}}th roots]] and [[zero of a function|roots of polynomials]] also allowed.
* An [[elementary function]]<ref group=note>Here "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree.</ref> is the same, with [[logarithm]]s and [[exponential functions]] allowed.