The rational function is .

First graph the function.

The rational function is .

Factor the numerator and denominator of . Find the domain of the rational function :

The domain of a rational function is the set of all real numbers except those for which the denominator is .

To find which number make the fraction undefined create an equation where the denominator is not equal to .

and

and .

The domain of function is .

Write in lowest terms :

The rational function .

The function is in lowest terms.  

Locate the intercepts of the graph and determine the behavior of the graph of near each - intercept :

Change to .

Find the intercepts.

To find -intercept equate the numerator to zero.

.

-intercept is .

Determine the behaviour of the graph of near each -intercept.

Near : .

Plot the point and indicate a line with negative slope.

Find the intercept by substituting in the rational function.

- intercept is .

Determine the vertical asymptotes :

Vertical asymptote can be found by making denominator.

or

or

Determine the horizantal asymptotes / oblique asymptotes :

To find horizontal asymptote, first find the degree of the numerator and the degree of denominator.

Degree of numerator, Degree of the denominator

Since the degree of the numerator is less than the degree of denominator,

Horizontal asymptote is .

Use the zeros of the numerator and denominator of to divide the -axis into intervals :

The real zeros of numerator is and the real zeros of denominator are and .

So the real zeros are divide the - axis into four intervals.

Choosing a number for in each interval and evaluating . 

End behavior of the graph :

and , hence the graph of is will approach the vertical asymptote, at .

and , hence the graph of is will approach the vertical asymptote,at  .

and , hence the graph of is will approach the horizontal asymptote, at  . 

Use the results obtained in Steps 1 through 7 to graph the function :

Draw the coordinate plane.

Next dash the horizontal and vertical asymptotes.

Plot the , intercepts and coordinate pairs found in the table.

Connect the plotted points.

When you draw your graph, use smooth curves complete the graph.

First determine the intervals of -such that the graph is below the - axis from the graph.

The graph of the function is below the - axis on the intervals or

From the graph,  for .

The solution set is  or in interval notation, .

; .