For example: $f\left( x \right) = \frac{{{x^3} + 5{x^2} - 8x + 6}}{{{x^2} - 1}}$ ; $g\left( x \right) = \frac{1}{{{x^2} - 4}}$ ; $h\left( x \right) = \frac{{ - 8x + 3}}{{2x - 5}}$

# What is a rational function?

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**Investigating Asymptotes on the Graph of Rational Functions**

Consider the rational function $f\left( x \right) = \frac{1}{{x - 2}}$ .- 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

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

- What is the non-permissible value of the rational function?
- 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

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

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- 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

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

- 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

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

- Complete the table of values below, then plot the points on the grid.