function

(noun)

a relation in which each element of the domain is associated with exactly one element of the co-domain

Related Terms

Examples of function in the following topics:

  • 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: $f(x)=ax^2+bx+c, a \ne 0$.

  • 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.

  • Further Transcendental Functions

    • A transcendental function is a function that is not algebraic.
    • Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
    • A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
    • Because of this, transcendental functions can be an easy-to-spot source of dimensional errors.
    • Bottom panel: Graph of sine function versus angle.

  • Inverse Functions

    • An inverse function is a function that undoes another function.
    • Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range $Y$, in which case the inverse relation is the inverse function.
    • Not all functions have an inverse.
    • Let's take the function $y=x^2+2$.
    • We can check to see if this inverse "undoes" the original function by plugging that function in for $x$:

  • The Derivative as a Function

    • If every point of a function has a derivative, there is a derivative function sending the point $a$ to the derivative of $f$ at $x = a$: $f'(a)$.
    • This function is written $f'(x)$ and is called the derivative function or the derivative of $f$.
    • Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions.
    • It is only defined on functions:
    • At the point where the function makes a jump, the derivative of the function does not exist.

  • Derivatives of Exponential Functions

    • The derivative of the exponential function is equal to the value of the function.
    • Functions of the form $ce^x$ for constant $c$ are the only functions with this property.
    • The slope of the graph at any point is the height of the function at that point.
    • The rate of increase of the function at $x$ is equal to the value of the function at $x$.
    • Graph of the exponential function illustrating that its derivative is equal to the value of the function.

  • Inverse Trigonometric Functions: Differentiation and Integration

    • The inverse trigonometric functions are also known as the "arc functions".
    • There are three common notations for inverse trigonometric functions.
    • The differentiation of trigonometric functions is the mathematical process of finding the rate at which a trigonometric function changes with respect to a variable.
    • Thus each function has an infinite number of antiderivatives.
    • Note that some of these functions are not valid for a range of $x$ which would end up making the function undefined.

  • Vector-Valued Functions

    • Also called vector functions, vector valued functions allow you to express the position of a point in multiple dimensions within a single function.
    • A three-dimensional vector valued function requires three functions, one for each dimension.
    • This is a three dimensional vector valued function.
    • This can be broken down into three separate functions called component functions:
    • This function is representing a position.