Rational function

Rational role of grade 2:

and=x2− − 3x− − 2x2− − 4{displaystyle y={cfrac {x^{2}-3x-2}{x^{2}-4}}}}}

Rational function of grade 3:

and=x3− − 2x2(x2− − 5){displaystyle y={cfrac {x^{3}-2x}{2(x^{2}-5)}}}}}

In mathematics, a rational function of a variable is a function that can be expressed in the form:

f(x)=P(x)Q(x){displaystyle f(x)={frac {P(x)}{Q(x)}}}

where P and Q are polynomials in the variable x{displaystyle x}and being Q different from the null polynomial, this fraction is irreducible, that is, the equations P(x) = 0 and Q(x) = 0 lack common roots. This definition can be extended to a finite but arbitrary number of variables, using polynomials of several variables:

f(x1,...... ,xn)=P(x1,...... ,xn)Q(x1,...... ,xn){displaystyle f(x_{1},dotsx_{n})={frac {P(x_{1},dotsx_{n})}{Q(x_{1},dotsx_{n}}}}}}}}}

The word "rational" it refers to the fact that the rational function is a ratio or quotient (of two polynomials); the coefficients of the polynomials can be rational numbers.

Rational functions have various applications in the field of numerical analysis to interpolate or approximate the results of other more complex functions, since they are computationally simple to calculate like polynomials, but allow a greater variety of behaviors to be expressed.

Analysis of rational functions

Zeros

A rational function is 0 only if your numerator is 0. It could be said, the roots of a rational function f(x)=P(x)Q(x){displaystyle {frac {P(x)}{Q(x)}}}}} are the roots of the polynomial of the P(x) numerator. For example, the function f(x)=x2− − 1x+3{displaystyle f(x)={frac {x^{2}-1}{x+3}} defined in all real numbers except -3, has as roots to the values where x2− − 1=0{displaystyle x^{2}-1=0}This is x=1 and x=-1.

Vertical Asymptotes

Vertical asymptotes are straight in the form x=a. In a rational function, vertical asymptotes are determined by the values that cancel the denominator. For example, the function f(x)=x2− − 1x+3{displaystyle f(x)={frac {x^{2}-1}{x+3}} has a vertical equation asymptote x=-3.

Horizontal Asymptotes

Horizontal asymptotes are lines of the form y=b. In a rational function, the horizontal asymptotes are determined as follows:

Sea f(x)=P(x)Q(x){displaystyle {frac {P(x)}{Q(x)}}}}} being P(x) and Q(x) polynomials and Q(x) is not null. We will call "gr(A)" to the degree of a polynomial A.

• If gr(Q) visgr(P) then f(x) does not have a horizontal asympt.
• If gr(Q) purgr(P) then f(x) has a horizontal asympt with equation and=0.
• If gr(P)=gr(Q) then f(x) has a horizontal asymp with equation and= CorecifientePrincipaldePCorecifientePrincipaldeQ{displaystyle {frac {CoephantPrincipaldeP}{CoeignPrincipaldeQ}}}}.

For example, the function f(x)=x2− − 1x+3{displaystyle f(x)={frac {x^{2}-1}{x+3}} does not have horizontal asymptote, since the degree of the polynomial of the numerator than the degree of the denominator.

Homographic function

A homographic function is a type of rational function, where the numerator is a polynomial of degree less than or equal to 1 and the denominator is a polynomial of degree one. That is, a function of the form:

f(x)=ax+bcx+d{displaystyle f(x)={frac {ax+b}{cx+d}}}}

where a, b, c, d are real numbers and c is not zero. If the denominator is different from zero and if ad ≠ bc, its graph corresponds to a hyperbola.. Applying the analysis of rational functions, the same can be done for homographic functions.

Zeros

A homographic function is 0 only if your numerator is 0. If a homographic function we write it as f(x)=ax+bcx+d{displaystyle f(x)={frac {ax+b}{cx+d}}}}then its root is determined by the root of ax+b. See ax+b=0 if x= -b/a. Then the root of a homographic function is x= -b/a. Note that if a=0, then the rational function has no roots.

Vertical Asymptotes

Be the function f(x)=ax+bcx+d{displaystyle f(x)={frac {ax+b}{cx+d}}}} vertical asymptotes are determined by the values that annul the denominator, that is, those that annul cx+d. As the denominator is a linear function, there is only one value that overrides it, and it is x=-d/c.

Horizontal Asymptotes

Be the function f(x)=ax+bcx+d{displaystyle f(x)={frac {ax+b}{cx+d}}}} with "a" not null, then the function has a horizontal equation asymptote and=a/c which is the quotient between the main coefficient of the numerator and the main coefficient of the denominator. If a=0, the function has a horizontal equation asymptote and=0, since the polynomial degree of the denominator is greater than that of the numerator.

Standard form

Another way in which homographic functions can be presented is as follows:

f(x)=Ax− − C+B{displaystyle f(x)={frac {A}{x-C}}}+B} where A,B,C are real numbers and A is not zero.

From that expression, it is easier to read the elements of the function to make its graph. Let's do a preliminary reading of that expression with what we already know.

Function f(x)=Ax− − C{displaystyle f(x)={frac {A}{x-C}}}}} has as R-{C} domain therefore has vertical asymptote in x=C. At the same time, the degree of the denominator's polynomial is grade 1, and the denominator's degree is 0 for being a constate. We are in the case where the horizontal asymptote is and=0.

When to f(x)=Ax− − C{displaystyle f(x)={frac {A}{x-C}}}}} let us add B, this will shift to the function vertically (up if B is positive and down if B is negative), therefore it will also shift to its horizontal asymptote. The vertical asymptote will be maintained as the domain remains the same when we add B. Therefore, we will always have horizontal asymptote in and = vertical asymptote in x=C.

In this case, if the function has a horizontal asymptote at y=B, we have that Im(f)=R-{B} since the function never touches the asymptote (“y” takes the value B).

Case B=0

If B=0 then the equation of the horizontal asymptote is y=0, therefore the function approaches the x axis without touching it. It makes sense to think then that the function has no roots, since f(x)=Ax− − C{displaystyle f(x)={frac {A}{x-C}}}}} (B=0) never worth 0 because Aiera0.

The function is positive when A and (x-C) have the same sign. So:

If A is a positive number, then f is positive when x>C. If A is a negative number f is positive when x < C.

If A is a negative number, f is positive when x<C. If A is a negative number f is positive when x>C.

Case C=0 and B≠0

If C=0, f(x)=Ax+B{displaystyle f(x)={frac {A}{x}} then the horizontal asymptote is not the x but y=B axis, so it is above or below the axis, the vertical asymptote is x=0.

Properties

• Any rational function is class C∞ ∞ {displaystyle C^{infty}} in a domain that does not include the roots of polynomial Q(x).
• All rational functions in which the degree of Q is greater or equal to the degree of P have asymptotes (vertical, horizontal or oblique).
• All rational functions whose coefficients belong to a body form a body that includes the base body as a subbody. The body of rational functions forms a subbody of the body of series of formal powers.

Integration of rational functions

Given a rational function:

f(x)=P(x)Q(x),P(x),Q(x)한 한 R[chuckles]x]{displaystyle f(x)={frac {P(x)}{Q(x)}}},qquad P(x),Q(x)in mathbb {R} [x]}}}

If the denominator is a monic polynomic Q(x){displaystyle scriptstyle Q(x)} with k different roots, then it will admit the following factorization in terms of irreducible polynomial:

{Q(x)=(x− − r1)m1(x− − r2)m2...... (x− − rk)mk(x2+s1x+t1)n1...... (x2+slx+tl)nlk,l,mi,nj한 한 N,rp,sp,tp한 한 R{displaystyle {begin{cases}Q(x)=(x-r_{1})^{m_{1}{x-r_{2}{m_{2}}{s}{x-r_{k}{s}{s}{s}{s}{s}{s}, }{s, }{s, }x

Yeah. <math alttext="{displaystyle scriptstyle {mbox{gr}}(P)gr(P).gr(Q){displaystyle scriptstyle {mbox{gr}}{mbox{gr}(Q)}<img alt="scriptstyle {mbox{gr}}(P) then the rational function can be written as a linear combination of rational fractions of the forms:

f1(x)=1(x− − ri)f2(x)=1(x− − ri)uf3(x)=1x2+a2f4(x)=1(x2+a2)vf5(x)=xx2+a2f6(x)=x(x2+a2)w{cHFFFFFF}{cH00FFFF}{cHFFFFFF}{cHFFFFFF}{cH00FFFF}{cH00FFFF}{cH00FFFF}{cHFFFFFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cH

So the integral of the function fi(x){displaystyle scriptstyle f_{i}(x)} is a linear combination of shape functions Fi(x){displaystyle scriptstyle F_{i}(x)}}:

F1(x)=ln (x− − ri)F2(x)=1− − u(x− − ri)u− − 1F3(x)=1aarctan xaF4(x)=12a2(x(v− − 1)(x2+a2)v− − 1+2v− − 3v− − 1∫ ∫ dx(x2+a2)v− − 1)F5(x)=12ln (x2+a2)F6(x)=− − 12(w− − 1)(x2+a2)w− − 1{cHFFFFFF}{cH00FF00}{cH00FF00}{cH00FF00}{cH00FF00}{cH00FF00}{cH00FF00}{cH00FFFF00}{cH00FF00}{cH00FFFF00}{cH00FFFFFF00}{cH00}{cH00FFFFFFFFFFFF00}{cH00}{cH00}{cH00FFFFFFFFFFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00}{cH00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00}{cH00}{cH00FF00FFFF00}{cH00}{cH00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

Note that the above implies that the rational functions constitute an algebraic field that is closed under derivation, but not under integration.