# What is a rational function?

### What is a rational function?

#### Lessons

A rational function is defined as a "ratio" of polynomials: $rational\;function = \frac{{polynomial}}{{polynomial}}$
For example: $f\left( x \right) = \frac{{{x^3} + 5{x^2} - 8x + 6}}{{{x^2} - 1}}$ ; $g\left( x \right) = \frac{1}{{{x^2} - 4}}$ ; $h\left( x \right) = \frac{{ - 8x + 3}}{{2x - 5}}$
• 1.
Investigating Asymptotes on the Graph of Rational Functions
Consider the rational function $f\left( x \right) = \frac{1}{{x - 2}}$ .
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 non-permissible value of the rational function?

c)
Now, let’s investigate the behaviour of the rational function near the non-permissible value by plotting more points close to the non-permissible 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 right-end 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)
To investigate the left-end behaviour of the rational function (as $x \to - \infty$), complete the table of values below and plot the points.
 $x$ -10 -100 -1000 $y = f\left( x \right) = \frac{1}{{x - 2}}$