Question

domain, range of a function, (x^2+1)/6x^2 + 13x - 6.

Answer 1

The rational function (x ² + 1 )/(6x ² + 13x - 6 )

We know that all possible values of x  is domain of a function.

A rational function is simply fraction and in a fraction the denominator cannot be equal to 0 because it would be undefined.

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

The quadratic form is 

So the domain of the function all real numbers except  .

 

Answer 2

(1)  The rational function  (x ² + 1 )/(6x ² + 13x - 6 )

The graph of rational functions can be recognised by the fact two or more parts.

To find  y intercept  x = 0 in the rational function.

y = (0+1)/(6(0 ²) +13(0)-6)

y = -1/6

In this case y  intercept is  -1/6.

2) to find x intercept let the numarator = 0

x ² + 1= 0

x ²  = -1

x = √(-1)

x = ± i

There is complex roots.

3) to find the vertical asympotote.

To find vertical asypototes are found  bysolving denominator = 0

Vertical asymptotes are

4) To find horizontal asymptote.

Degree of the numarator = 2 and the degree of denominator = 2.

If the degree of the numerator is equal to the degree of the denominator,

leading coefficient of numarator is 1 and leading coefficient of denominator is 6

y = 1/6 is the horizontal asymptote.

Now pick few more x - values, compute the corresponding y  values and flat few more points.

x	y = (x 2 + 1 )/(6x 2 + 13x - 6 )	(x , y )
-1	y = (-1 2 + 1 )/(6(-1) 2 + 13(-1) - 6 ) = 2/-13 =-0.15	(-1,-0.15)
-5	y = (-5 2 + 1 )/(6(-5) 2 + 13(-5) - 6 ) = 26/79=0.32	(-5, 0.32)
1	y = (1 2 + 1 )/(6(1) 2 + 13(1) - 6 ) = 2/13 = 0.15	(1,0.15)
5	y = (5 2 + 1 )/(6(5) 2 + 13(5) - 6 ) = 26/209=0.12	(5,0.12)

Graph

1) Draw the coordinate plane.

2) Next dash the horizontal and vertical asympototes.

3) Plot the y intercept and coordinate pairs found in the table..

4) Connect the plotted points .

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

From the graph range set is the corresponding values of the function for different values of x.

Range

.

Domain set.