Tool to check the parity of a function (even or odd functions): it defines the ability of the function (its curve) to verify symmetrical relations.

Even or Odd Function - dCode

Tag(s) : Functions

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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 $. Odd functions exhibit rotational symmetry of 180 degrees, with their graphs rotating by 180 degrees about the origin.

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

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 $.

Polynomials of even degree are generally even functions.

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.

NB: An odd function cancels $ f(x)=0 $ necessarily in $ x=0 $

__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 $.

Polynomials of odd degree are generally odd functions.

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.

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.

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

Every even function has a vertical axis of symmetry: the ordinate axis $ y $.

Any odd function has a central symmetry with center at the origin (0,0).

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Cite as source (bibliography):

*Even or Odd Function* on dCode.fr [online website], retrieved on 2024-10-05,

- Even and Odd Function Calculator
- What is the parity of a function? (Definition)
- How to check if a function is even?
- How to check if a function is odd?
- How to check if a function is neither even nor odd?
- What is the parity of trigonometric functions (cos, sin, tan)?
- Why are functions called even or odd?
- Is there a function that is both even and odd?
- How to complete the graph of an even (or odd) function?

even,odd,function,parity,symmetry,trigonometric,cosine,sine

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