A function of a function – Wikipedia

In mathematics, the parity of a function of a real, complex or vector variable is a property which first requires the symmetry of the definition field in relation to the origin, then is expressed by one or the other of the following relations:

• pair function : for everything x of the field of definition, f ( – x ) = f ( x ) ;
• unusual function : for everything x of the field of definition, f ( – x ) = – f ( x ) .

In actual analysis, pairs functions are the functions whose representative curve is symmetrical with respect to the axis of orders, such as constant functions, square function and more generally the functions ^{[ first ]} , the Cosinus and Cosinus hyperbolic functions…
The unclean functions are those whose representative curve is symmetrical with respect to the origin, such as identity, cube functions and more generally the powers of omnipotence, reverse, sinus, tangent, hyperbolic and hyperbolic -tangent functions and their reciprocal.

The only functions to be both pairs and odd are zero functions on a symmetrical domain.

Any function is generally neither pair nor odd, even if his field of definition is symmetrical compared to the origin. Any function defined on such a field is uniquely written as a sum of a pair function and an odd function.

The highlighting of the parity of a function of a real variable (whether pair or odd) makes it possible to limit its study to the positive realities.

The parity of the functions serves, for example, to study the functions only on half of their definition interval, the other half being deduced by symmetry. It will be noted that an odd function, defined in 0, is zero at this point (indeed, since

${displaystyle f}$

is odd,

${displaystyle f(-x)=-f(x)}$

for everything

${displaystyle x}$

, and so

${displaystyle f(0)=-f(0)}$

; Thus

${displaystyle f(0)=0}$

.

We can also simplify the full calculation in the case of pair or odd function, since

${displaystyle int _{-n}^{n}f(x),mathrm {d} x}$

For

${displaystyle f}$

pair is equal to

${displaystyle 2times int _{0}^{n}f(x)dx,}$

, what we visualize well with the graphic representation of the area under the curve, and respectively

${displaystyle int _{-n}^{n}f(x),mathrm {d} x,}$

For

${displaystyle f}$

odd is equal to

${Displaystyle 0}$

. Indeed, there will be such a large positive area of

${Displaystyle 0}$

To

${displaystyle n}$

that of negative area of

${displaystyle -n}$

To

${Displaystyle 0}$

.

This definition of parity and imparity can also be explained with the notion of symmetrized of a function: the symmetrized function of a function s is the function with associated with

${displaystyle s(-x)}$

to a

${displaystyle x}$

given and, for example, s is pair if it is equal to its symmetry.

Pair part and odd parts of a function [ modifier | Modifier and code ]

And

${displaystyle E}$

is a subset of

${displaystyle mathbb {R} }$

symmetrical with respect to 0 (that is to say that if

${displaystyle x}$

belongs to

${displaystyle E}$

SO

${displaystyle -x}$

belongs to

${displaystyle E}$

), any function

${displaystyle f:Eto mathbb {R} }$

Can be uniquely decomposed as a sum of a pair function and an odd function:

${displaystyle f(x)=f_{text{p}}(x)+f_{text{i}}(x)}$

,

where the pair function is

${displaystyle f_{text{p}}(x)={frac {f(x)+f(-x)}{2}}}$

and the odd function is

${displaystyle f_{text{i}}(x)={frac {f(x)-f(-x)}{2}}}$

.

In effect ^{[ 2 ]} , the vector subspace of pairs functions and that of unclean functions are additional in the space of the functions of

${displaystyle E}$

In

${displaystyle mathbb {R} }$

.

Therefore, we can talk about the pair of

${displaystyle f}$

and its unclean part. For example, the exponential function breaks down as the sum of the hyperbolic cosine functions,

${displaystyle xmapsto {frac {mathrm {e} ^{x}+mathrm {e} ^{-x}}{2}}}$

and hyperbolic sinus,

${displaystyle xmapsto {frac {mathrm {e} ^{x}-mathrm {e} ^{-x}}{2}}}$

.

Either

${displaystyle f}$

A function defined on

${displaystyle E}$

And

${displaystyle (C_{f})}$

its graph, in a benchmark of axes

${displaystyle (Ox),(Oy)}$

.

• 

  Function

  ${displaystyle xmapsto x^{4}}$

  is pair.

• 

  Function

  ${displaystyle xmapsto x^{3}-3x}$

  is odd.

But a function whose representative curve has an axis or center of symmetry is not necessarily pair or odd: it is necessary that the center is

${displaystyle O}$

or the axis either

${displaystyle (Oy)}$

.

• Any constant function is pair.
• Any pair and monotonous function on its definition set is constant.
• The only function that is both pair and odd is zero function (constant function equal to zero).
• In general, the sum of a pair function and an odd function is neither pair nor odd; ex :
  ${displaystyle xmapsto 1+x}$

  .

• The sum or the difference in two pairs functions is pair.
• The sum or the difference in two odd functions is odd.
• Parity follows, for the product or the quotient, the signs rule: any product or quotient of two pairs functions is a pair, any product or quotient of two odd functions is also a pair, any product or quotient of a Pair function by an odd function is an unclean function.
• The derivative of a pair function is an odd function; The derivative of an odd function is a pair function.
• A primitive of an odd function on AND is not necessarily pair, unless AND is an interval.
• A primitive of a pair function on AND is not necessarily odd, unless AND is an interval and if the primitive considered is the one that cancels in 0.
• The composed of two odd functions is odd; The compound g ∘ f of a pair function g with an unusual function f is a pair function.
• The compound g ∘ f of any function g with a pair function f is a pair function.