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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

## Even and Odd Function Calculator

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.

### 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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