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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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 $. 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 $
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 $.
Polynomials of even degree are generally even functions.
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.
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.
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 $
How to complete the graph of an even (or odd) function?
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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Even or Odd Function on dCode.fr [online website], retrieved on 2024-11-21,
- 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?
