what is the domain and range of the function h(x), defined as h(x)= x+4/x^2-5x

Question

step by step guide.

Answer 1

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.

So the domain of the function all real numbers except  0 and 5.

(1)  The rational function  h (x ) = (x + 4)/(x ² - 5x )

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

1) y = (x + 4) / (x ² - 5x )

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

y = (0+4)/(0 ² - 5(0)

not defined.

In this case y  intercept is none.

2) to find x intercept let the numarator = 0

x + 4 = 0

x  = -4

3) to find the vertical asympotote.

To find vertical asypototes are found  bysolving denominator = 0

x ² - 5x = 0

x (x - 5) = 0

x = 0 and x - 5 = 0

Vertical asymptotes are x  = 0 and x  = 5.

4) To find horizontal asymptote.

Degree of the numarator = 1 and the degree of denominator = 2.

If the degree of the numerator is less than the degree of the denominator, the line

y = 0 (x - axis) is the horizontal asymptote.

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

x	y = (x + 4)/(x 2 - 5x )	(x , y )
-1	y = (-1 + 4) / (-1 2 -5(-1)) = 3/4	(-1,3/4)
-5	y = (-5 + 4) / (-5 2 -5(-5)) = -1/50	(-5, -1/50)
1	y = (1 + 4) / (1 2 -5(1)) = 5/-4	(1,-4/5)
5	y = (5 + 4) / (5 2 -5(5)) = 9/0	not defined
10	y = (10 + 4) / (10 2 -5(10)) = 14/50	(10,14/50)

Graph

1) Draw the coordinate plane.

2) Next dash the horizontal and vertical asympototes

3) Plot the x 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

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Domain set

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