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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 -

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# Even or Odd Function

## Even and Odd Function Calculator

### 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.

An even function will provide an identical image for opposite values.

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.

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 $f$.

Having proved 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?

Definition: 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.

Example: Determine 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.

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: $f(0) = 0$

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 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 $f(x) = -f(-x) = 0$

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