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Rational function is defined in similar fashion as rational number is defined in terms of numerator and denominator. Implicitly, we refer “real” rational function here. It is defined as the ratio of two real polynomials with the condition that polynomial in the denominator is not a zero polynomial.
$$f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)};\phantom{\rule{1em}{0ex}}q\left(x\right)\ne 0$$
Rational function is not defined for values of x for which denominator polynomial evaluates to zero as ratio “p(x)/0” is not defined. Some examples of rational function are :
$$f\left(x\right)=\frac{2{x}^{2}-x+1}{2{x}^{2}-5x-3};\phantom{\rule{1em}{0ex}}x\ne -\frac{1}{2},x\ne 3$$
$$g\left(x\right)=\frac{x+1}{2{x}^{2}-x+1}$$
$$h\left(x\right)=\frac{2{x}^{4}-{x}^{2}+1}{x+1};\phantom{\rule{1em}{0ex}}x\ne -1$$
Note second example function, g(x) above. There is no exclusion point for this rational polynomial. The denominator polynomial is $2{x}^{2}-x+1$ , whose determinant is negative and coefficient of ${x}^{2}$ term is positive. It means denominator of g(x) is positive for all values of x. We should also note that values of x being excluded are points - not a continuous interval. Further, the notation to denote exclusion is an “inequation” – not “inequality” – because notation $x\ne -1$ negates corresponding equation x = -1. Recall that inequality, on the other hand, compares relative values.
Domain of rational function is domain of numerator polynomial minus exclusion points as determined by zeroes of denominator polynomial. Since domain of polynomial is R, domain of rational polynomial is R minus exclusion points determined by denominator. The domains for three rational functions given above are :
$$\text{Domain of f(x)}=R-\{-\frac{1}{2},3\}$$ $$\text{Domain of g(x)}=R$$ $$\text{Domain of h(x)}=R-\{-1\}$$
Important properties are :
Singularities are x-values for which denominator of rational function is zero. The function is not defined for such x-values and as such these values are excluded from the domain set of the function. These points are also called exception points. Function is not defined at these points.
Factorizing numerator and denominator of rational function helps to identify singularities of algebraic rational function. Singularities correspond to x values resulting from equating linear factors in denominator to zero. The important thing to note here is that singularity or exception occurs when denominator of rational function turns zero – no matter whether linear factor in the denominator cancels out with the linear factor in numerator or not. To understand this point, let us consider few rational functions given below :
$$f\left(x\right)=\frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+1\right)}$$ $$g\left(x\right)=\frac{{\left(x-1\right)}^{2}\left(x+2\right)}{\left(x-1\right)\left(x+1\right)}$$ $$h\left(x\right)=\frac{\left(x-1\right)\left(x+2\right)}{{\left(x-1\right)}^{2}\left(x+1\right)}$$
We can see that h(x) contains a linear factor (x-1) in the denominator after cancellation of like linear factors. On the other hand, functions f(x) and g(x) do not contain (x-1) in the denominator after cancellation of like linear factors. The function g(x), however, contains (x-1) in the numerator after cancellation. Notwithstanding these possibilities, denominator of the rational function turns zero at x=1. As such, the point specified by x=1 is singularity for all three function forms shown above.
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