5.1 What is a rational function?
In previous chapter, we had an introduction about Rational Expressions. These are expressions which are mathematically defined as any expression that is in the form of p/q where q is any number that is not equal to zero. In the first part of this chapter, we will talk about the Rational Functions which are functions expressed as the ratio of two polynomials, p/q, where q is never equal to zero. We will get to understand more about them by giving more examples. In some examples, we would see rational functions.
In the second part of this chapter we will learn all about the point of discontinuity. The point of discontinuity will be the value of the x that would make the function irrational. Graphically this will be reflected as the point where the graph forms a break or holes.
The discontinuity formed between a break between two graphs would be referred to as an infinitive Discontinuity or Vertical asymptote. As we have learned in the previous lessons, asymptotes like the vertical, horizontal or slant asymptotes are breaks or spaces between two curves in a graph. In the third and fourth part of this chapter we will discuss all about the vertical and horizontal asymptotes.
We will also learn in those two chapters about the nonpermissible values. These are simply the values that would make the numerator and denominator irrational. In order for us to solve for the non permissible values, we have to solve for the common factor to get the value of x that would make the function irrational.
In the last part of the chapter we will learn how to graph rational function. We will also get to see and understand the end behaviour of the rational functions.
At the end of the chapter, we are expected to have a better understanding of rational functions which is needed for the proceeding chapters.
What is a rational function?
Lessons
Notes:
A rational function is defined as a "ratio" of polynomials:
$rational\;function = \frac{{polynomial}}{{polynomial}}$
For example: $f\left( x \right) = \frac{{{x^3} + 5{x^2}  8x + 6}}{{{x^2}  1}}$ ; $g\left( x \right) = \frac{1}{{{x^2}  4}}$ ; $h\left( x \right) = \frac{{  8x + 3}}{{2x  5}}$

a)
Complete the table of values below, then plot the points on the grid.
$x$
5
4
3
2
1
0
1
2
3
4
5
$y = f\left( x \right) = \frac{1}{{x  2}}$

b)
What is the nonpermissible value of the rational function?

c)
c) Now, let’s investigate the behaviour of the rational function near the nonpermissible value by plotting more points close to the nonpermissible value.
$x$
1.5
1.9
1.99
2
2.01
2.1
2.5
$y = f\left( x \right) = \frac{1}{{x  2}}$
undefined

d)
To investigate the rightend behaviour of the rational function (as $x \to \infty$), complete the table of values below and plot the points.
$x$
10
100
1000
$y = f\left( x \right) = \frac{1}{{x  2}}$

e)
e) To investigate the leftend behaviour of the rational function (as $x \to  \infty$), complete the table of values below and plot the points.
$x$
10
100
1000
$y = f\left( x \right) = \frac{1}{{x  2}}$
