Determine whether the function is a polynomial function calculator

1. Mathematics
2. Functions
3. Even or Odd Function

Even and Odd Function Calculator

Answers to Questions (FAQ)

What is the parity of a function? (Definition)

The parity of a function is a property giving the curve of the function characteristics of symmetry (axial or central).

— A function is even if the equality $$ f(x) = f(-x) $$ is true for all $ x $ from the domain of definition. An even function will provide an identical image for opposite values. Graphically, this involves that opposed abscissae have the same ordinates, this means that the ordinate y-axis is an axis of symmetry of the curve representing $ f $.

— A function is odd if the equality $$ f(x) = -f(-x) $$ is true for all $ x $ from the domain of definition. An odd function will provide an opposite image for opposite values. Graphically, this involves that opposed abscissae have opposed ordinates, this means that the origin (central point) (0,0) is a symmetry center of the curve representing $ f $.

NB: if an odd function is defined in 0, then the curve passes at the origin: $ f(0) = 0 $

How to check if a function is even?

To determine/show that a function is even, check the equality $ f(x) = f(-x) $, if the formula is true then the function is even.

Example: Determine whether the function is even or odd: $ f(x) = x^2 $ (square function) in $ \mathbb{R} $, the calculation is $ f(-x) = (-x)^2 = x^2 = f(x) $, so the square function $ f(x) $ is even.

Studying/Proving this equality for a single value like $ f(1) = f(-1) $ does not allow to conclude that there is parity, only to say that 1 and -1 have the same image by the function $ f $.

How to check if a function is odd?

To determine/tell that a function is odd, check the equality $ f(x) = -f(-x) $, if the formula is true then the function is even.

Example: Study whether the function is even or odd: $ f(x) = x^3 $ (cube function) in $ \mathbb{R} $, the calculation is $ -f(-x) = -(-x)^3 = x^3 = f(x) $, so the cube function $ f(x) $ is odd.

Having proved equality for a single value like $ f(2) = -f(-2) $ does not allow us to conclude that there is imparity, only to say that 2 and -2 have opposite images by the function $ f $.

How to check if a function is neither even nor odd?

A function is neither odd nor even if neither of the above two equalities are true, that is to say: $$ f(x) \neq f(-x) $$ and $$ f(x) \neq -f(-x) $$

Example: Determine the parity of $ f(x) = x/(x+1) $, first calculation: $ f(-x) = -x/(-x+1) = x/(x-1) \neq f(x) $ and second calculation: $ -f(-x) = -(-x/(-x+1)) = -x/(x-1) = x/(-x+1) \neq f(x) $ therefore the function $ f $ is neither even nor odd.

What is the parity of trigonometric functions (cos, sin, tan)?

In trigonometry, the functions are often symmetrical:

The cosine function $ \cos(x) $ is even.

The sine function $ \sin(x) $ is odd.

The tangent function $ \tan(x) $ is odd.

Why are functions called even or odd?

Developments in convergent power series or polynomials of even (respectively odd) functions have even degrees (respectively odd).

Is there a function that is both even and odd?

Yes, the function $ f(x) = 0 $ (constant zero function) is both even and odd because it respects the 2 equalities $ f(x) = f(-x) = 0 $ and $ f(x) = -f(-x) = 0 $

Source code

dCode retains ownership of the "Even or Odd Function" source code. Except explicit open source licence (indicated Creative Commons / free), the "Even or Odd Function" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Even or Odd Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Even or Odd Function" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

Cite dCode

The copy-paste of the page "Even or Odd Function" or any of its results, is allowed as long as you cite dCode!
Cite as source (bibliography):
Even or Odd Function on dCode.fr [online website], retrieved on 2022-09-30, https://www.dcode.fr/even-odd-function

Summary :

The degree function calculates online the degree of a polynomial.

degree online

Description :

The computer is able to calculate online the degree of a polynomial.

Calculating the degree of a polynomial

The calculator may be used to determine the degree of a polynomial.

To obtain the degree of a polynomial defined by the following expression `x^3+x^2+1`, enter : degree(`x^3+x^2+1`) after calculation, the result 3 is returned.

Calculating the degree of a polynomial with symbolic coefficients

The calculator is also able to calculate the degree of a polynomial that uses letters as coefficients.

To obtain the degree of a polynomial defined by the following expression : `ax^2+bx+c` enter degree(`ax^2+bx+c`) after calculation, result 2 is returned.

Syntax :

degree(polynomial)

Examples :

degree(`x^3+x^2+1`), returns 3

Calculate online with degree (degree of a polynomial)

Is it a polynomial or not?