Examples for

Mathematical Functions

In mathematics, a function is defined as a relation, numerical or symbolic, between a set of inputs (known as the function's domain) and a set of potential outputs (the function's codomain). The power of the Wolfram Language enables Wolfram|Alpha to compute properties both for generic functional forms input by the user and for hundreds of known special functions. Use our broad base of functionality to compute properties like periodicity, injectivity, parity, etc. for polynomial, elementary and other special functions.

Domain & Range

Compute the domain and range of a mathematical function.

Compute the domain of a function:

domain of f(x) = x/(x^2-1)

Compute the range of a function:

range of e^(-x^2)

Compute domain and range of a function of several variables:

domain and range of z = x^2 + y^2
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Continuity

Determine the continuity of a mathematical function.

Determine whether a function is continuous:

is sin(x-1.1)/(x-1.1)+heaviside(x) continuous

Locate discontinuities of a function:

discontinuities (x^3+8)/(x^3+3x^2-4x-12)
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Special Functions

Compute properties of multiple families of special functions.

Compute properties of a special function:

zeta(x)

Numerically evaluate a special function:

sn(2.1, 0.5)

Do computations with special functions:

d/dx Si(x)^2
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Injectivity & Surjectivity

Determine the injectivity and surjectivity of a mathematical function.

Determine whether a given function is injective:

is y=x^3+x a one-to-one function?

Determine whether a given function is surjective:

is x^2-x surjective?
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Periodic Functions

Compute the period of a periodic function.

Compute the period of a periodic function:

period y=sin(x)*cos(3x)

Find periods of a function of several variables:

period sin(x + y^2 - 3z)
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Number Theoretic Functions

Get information about arithmetic functions, such as the Euler totient and Möbius functions, and use them to compute properties of positive integers.

Get information about a number theoretic function:

Euler phi

Do computations with number theoretic functions:

phi(110)
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RELATED EXAMPLES

  • Calculus & Analysis
  • Continued Fractions
  • Differential Equations
  • Polynomials
  • Rational Functions

    • Even & Odd Functions

    Determine the parity of a mathematical function.

    Determine whether a function is even or odd:

    what is the parity of sin(x)
    is sin(x+pi/4)+cos(x+pi/4) an even function?
    is f(x,y,z)=sin(xyz) an odd function?
    Representation Formulas

    Compute alternative representations of a mathematical function.

    Find representations of a function of a given type:

    gamma(x) integral representation
    Si(x) limit rep
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