The rational function . 

Find the domain of the rational function :

The domain of a rational function is the set of all real numbers for which the function is mathematically correct.

Denominator of the function should not be zero.

and

and .

The domain of function is .

Write in lowest terms :

The function is .

.

The function is in lowest terms.

The rational function .

Change to .

Find the intercepts.

Find the -intercept by equating to zero.

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.

There is no -intercepts.

Vertical asymptote can be found by making denominator to zero.

or

or .

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 the denominator,

hence horizontal asymptote is .

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

So divide the - axis into four intervals.

Behavior of the graph :

As and , hence the graph of approaches to a vertical asymptote at.

As and , hence the graph of approaches to a vertical asymptote at .

As and , hence the graph of approaches to a horizontal asymptote at .

Graph :

Draw the coordinate plane.

Plot the horizontal and vertical asymptotes.

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

Connect the plotted points to smooth curve.

The domain of function is .

The rational function in lowest terms .

-intercept is .

There is no -intercepts.

Vertical asymptote are or .

Horizontal asymptote is .

Graph of the rational function . 

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