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Even or Odd Function

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.

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Even or Odd Function -

Tag(s) : Functions

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Even or Odd Function

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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.

Answers to Questions

How to check if a function is even?

Definition : A function is even if the equality $$ f(x) = f(-x) $$ is true for all x from the domain of definition.

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

Graphically, this involves that opposed abscissae have the same ordinates, this means that the ordinate axis is an axis of symmetry of the curve representing f.

How to check if a function is odd?

Definition : A function is odd if the equality $$ f(x) = -f(-x) $$ is true for all x from the domain of definition.

Example: $ f(x) = x^3 $ in $ \mathbb{R} $, le calculation is $ -f(-x) = -(-x)^3 = x^3 = f(x) $, so $ f(x) $ is odd.

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: $$ f(0) = 0 $$

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

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).

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