3.4 Rational function  (Page 2/6)

We shall see that either a hole or vertical asymptote occurs at the point of exception i.e. singularity. It depends on how do the linear factors in the denominator relate to linear factors in the numerator.

Holes

Hole exists at a singularity when corresponding linear factor of the denominator cancel out completely or when linear factor remains in the numerator after cancellation. Hole is a point on the graph where function value is not defined. It is a point having x and y coordinates. We determine x-coordinate by equating linear factor in denominator to zero. Its y-coordinate is obtained by plugging x-value in the reduced (after cancellation of common factors in numerator and denominator) function form. As pointed out, there are two situations with respect to position of a hole :

1: If linear factor cancels completely, then hole lies any where but not on x-axis.

$$f\left(x\right)=\frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+1\right)}$$

Here, singularities occur at x= -1 and 1. The linear factor (x-1) is present in both numerator and denominator and as such cancels out completely. Therefore, there is a hole at x=1. On the other hand, no cancellation is involved at x=-1. There exists a vertical asymptote at x=-1. Important point to realize here is that if linear factor is present in denominator only and there is no cancellation involved, then x-value corresponding to linear factor is not the x-coordinate of hole. We shall learn about vertical asymptote subsequently. Now, the y-coordinate of hole is :

$$\Rightarrow f\left(x\right)=\frac{\left(x+2\right)}{\left(x+1\right)}=\frac{3}{2}=1.5$$

The graph of function, f(x), is shown in the figure below :

2: If linear factor remains in the numerator after cancellation, then hole lies on x-axis. The graph tends to intercept x-axis. As such, hole exists at x-axis.

$$g\left(x\right)=\frac{{\left(x-1\right)}^2\left(x+2\right)}{\left(x-1\right)\left(x+1\right)}$$

Here, singularities occur at x= -1 and 1. There is a vertical asymptote at x=-1, but a hole at x=1. The y-coordinate of hole is :

$$\Rightarrow g\left(x\right)=\frac{\left(x-1\right)\left(x+2\right)}{\left(x+1\right)}=\frac{0X3}{2}=0$$

Thus, hole lies on x-axis.

Asymptotes

An asymptote is a straight line at singularity which graph of function tends to approach but never touches. The difference between graph and asymptotes is infinitesimally small as the graph is extended away from x-axis. Important to note is that graph neither touches or crosses the asymptote. In the case of vertical asymptote, the function values tend to be either positive or negative large number or a combination of two on either side of the vertical asymptote. In the case of horizontal asymptote, the function values tend to be a finite value.

Vertical asymptote

Vertical asymptote is a vertical line including y-axis to which graph of function comes closer and closer but never touches. Vertical asymptotes correspond to very large y-values, where difference between x-value and asymptote is infinitesimally small. An equation of vertical asymptote has the form,