function
Algebra
Any mathematical formula that produces one and only one result for each input.
A relation in which each element of the input is associated with exactly one element of the output.
Calculus
Examples of function in the following topics:

Functional Groups
 Functional groups are atoms or small groups of atoms (two to four) that exhibit a characteristic reactivity when treated with certain reagents.
 A particular functional group will almost always display its characteristic chemical behavior when it is present in a compound.
 Because of their importance in understanding organic chemistry, functional groups have characteristic names that often carry over in the naming of individual compounds incorporating specific groups.
 In the following table the atoms of each functional group are colored red and the characteristic IUPAC nomenclature suffix that denotes some (but not all) functional groups is also colored.

Inverse Trigonometric Functions
 Each trigonometric function has an inverse function that can be graphed.
 To use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse.
 However, the sine, cosine, and tangent functions are not onetoone functions.
 As with other functions that are not onetoone, we will need to restrict the domain of each function to yield a new function that is onetoone.
 The arcsine function is a reflection of the sine function about the line $y = x$.

Introduction to Rational Functions
 A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
 A rational function is any function which can be written as the ratio of two polynomial functions.
 Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
 Note that every polynomial function is a rational function with $Q(x) = 1$.
 A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.

Functional Structure
 An organization with a functional structure is divided based on functional areas, such as IT, finance, or marketing.
 Functional departments arguably permit greater operational efficiency because employees with shared skills and knowledge are grouped together by functions performed.
 Functional structures may also be susceptible to tunnel vision, with each function perceiving the organization only from within the frame of its own operation.
 This organizational chart shows a broad functional structure at FedEx.
 Each different functions (e.g., HR, finance, marketing) is managed from the top down via functional heads (the CFO, the CIO, various VPs, etc.).

Functional Groups
 Functional groups also play an important part in organic compound nomenclature; combining the names of the functional groups with the names of the parent alkanes provides a way to distinguish compounds.
 Functionalization refers to the addition of functional groups to a compound by chemical synthesis.
 In materials science, functionalization is employed to achieve desired surface properties; functional groups can also be used to covalently link functional molecules to the surfaces of chemical devices.
 Alcohols are a common functional group (OH).
 Define the term "functional group" as it applies to organic molecules

Expressing Functions as Power Functions
 A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
 A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
 Polynomials are made of power functions.
 Functions of the form $f(x) = x^3$, $f(x) = x^{1.2}$, $f(x) = x^{4}$ are all power functions.
 Describe the relationship between the power functions and infinitely differentiable functions

Linear and Quadratic Functions
 Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.
 They are one of the simplest functional forms.
 Linear functions may be confused with affine functions.
 However, the term "linear function" is quite often loosely used to include affine functions of the form $f(x)=mx+b$.
 A quadratic function, in mathematics, is a polynomial function of the form: $f(x)=ax^2+bx+c, a \ne 0$.

Increasing, Decreasing, and Constant Functions
 As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.
 We say that a function is increasing on an interval if the function values increase as the input values increase within that interval.
 In terms of a linear function $f(x)=mx+b$, if $m$ is positive, the function is increasing, if $m$ is negative, it is decreasing, and if $m$ is zero, the function is a constant function.
 In mathematics, a constant function is a function whose values do not vary, regardless of the input into the function.
 A function is a constant function if $f(x)=c$ for all values of $x$ and some constant $c$.

Inverse Functions
 An inverse function is a function that undoes another function: For a function $f(x)=y$ the inverse function, if it exists, is given as $g(y)= x$.
 Inverse function is a function that undoes another function: If an input $x$ into the function $f$ produces an output $y$, then putting $y$ into the inverse function $g$ produces the output $x$, and vice versa. i.e., $f(x)=y$, and $g(y)=x$.
 If $f$ is invertible, the function $g$ is unique; in other words, there is exactly one function $g$ satisfying this property (no more, no less).
 Not all functions have an inverse.
 A function $f$ and its inverse $f^{1}$.

Continuity
 A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.
 A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.
 Otherwise, a function is said to be a "discontinuous function."
 A continuous function with a continuous inverse function is called "bicontinuous."
 This function is continuous.