function

(noun)

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

Related Terms

  • secant line
  • derivative
  • inverse
  • arithmetic mean
  • average

Examples of function in the following topics:

  • Expressing Functions as Power Functions

    • A power function is a function of the form f(x)=cxrf(x) = cx^rf(x)=cx​r​​ where ccc and rrr are constant real numbers.
    • A power function is a function of the form f(x)=cxrf(x) = cx^rf(x)=cx​r​​ where ccc and rrr are constant real numbers.
    • Polynomials are made of power functions.
    • Functions of the form f(x)=x3f(x) = x^3f(x)=x​3​​, f(x)=x1.2f(x) = x^{1.2}f(x)=x​1.2​​, f(x)=x−4f(x) = x^{-4}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+bf(x)=mx+bf(x)=mx+b.
    • A quadratic function, in mathematics, is a polynomial function of the form: f(x)=ax2+bx+c,a≠0f(x)=ax^2+bx+c, a \ne 0f(x)=ax​2​​+bx+c,a≠0.
  • Inverse Functions

    • An inverse function is a function that undoes another function: For a function f(x)=yf(x)=yf(x)=y the inverse function, if it exists, is given as g(y)=xg(y)= xg(y)=x.
    • Inverse function is a function that undoes another function: If an input xxx into the function fff produces an output yyy, then putting yyy into the inverse function ggg produces the output xxx, and vice versa. i.e., f(x)=yf(x)=yf(x)=y, and g(y)=xg(y)=xg(y)=x.
    • If fff is invertible, the function ggg is unique; in other words, there is exactly one function ggg satisfying this property (no more, no less).
    • Not all functions have an inverse.
    • A function fff and its inverse f−1f^{-1}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 YYY, in which case the inverse relation is the inverse function.
    • Not all functions have an inverse.
    • Let's take the function y=x2+2y=x^2+2y=x​2​​+2.
    • We can check to see if this inverse "undoes" the original function by plugging that function in for xxx:
  • The Derivative as a Function

    • If every point of a function has a derivative, there is a derivative function sending the point aaa to the derivative of fff at x=ax = ax=a: f′(a)f'(a)f​′​​(a).
    • This function is written f′(x)f'(x)f​′​​(x) and is called the derivative function or the derivative of fff.
    • 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 cexce^xce​x​​ for constant ccc 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 xxx is equal to the value of the function at xxx.
    • 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 xxx 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.
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