what is the range of (-x^2 + 3x + 9)/(x-1)

Question

What is the domain and range of the function (-x^2 + 3x + 9)/(x-1)?

Answer 1

Given function f(x)= y = (-x^2+3x+9)/(x-1)

For x = 0

y = (-0+3*0+9)/(0-1) = -9

For x = 1

y = (-(1)^2+3*1+9)/(1-1)

y = (-1+3+9)/0 = Not defined

For x = 2

y = (-(2)^2+3*2+9)/(2-1)

y = (-4+6+9)/1= 11

For x = 3

y = (-(3)^2+3*3+9)/(3-1)

y = (-9+9+9)/2 = 9/2

For x = -1

y = (-(-1)^2+3*-1+9)/(-1-1)

y = (-1-3+9)/-2 = -5/2

For x = -2

y = (-(2)^2+3*-2+9)/(-2-1)

y = (-4-6+9)/-3 = 1/3

For x = -3

y = (-(-3)^2+3*-3+9)/(-3-1)

y = (-9-9+9)/-4 = 9/4

x value is half coordinate pair and y value is the half coordinate pair.

If x = -3, -2, -1, 0, 2, 3 respectively Substitute the x values in given function the set of coordinate pair is be 9/4,1/3, -2,-9,3,-1.

We know that x values are domain of the function and y values is range of the function.

Domain set is {-3,-2,-1,1,2,3}

Range set is {9/4,1/3,-2,-9,3,1}

Answer 2

The function f (x ) = (- x ² + 3x + 9)/(x - 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 .

Domain set is .

 

Answer 3

Continuous...

To determine the range, graph the rational function.

The rational function f (x ) = (- x ² + 3x + 9)/(x - 1)

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

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

y = (- 0 ² + 3(0) + 9)/(0 - 1)

y = 9

y  intercept is  (0,9).

ii) to find x intercept let the numarator = 0.

- x ² + 3x + 9 = 0

x = -3(-1+2.23)/2 and x = 3(1+2.23)/2

x = -1.85 and x = 4.84

x  intercepts are (-1.85,0) and (4.84,0)

iii) To find the vertical asympotote.

To find vertical asympototes are found  bysolving denominator = 0.

x  -1 = 0

x  = 1

Vertical asympotote is x  = 1.

In this case degree of the numarator = 2 and the degree of denominator = 1.

If the degree of the numarator is exactly one more than degree of the denominator.

The asympotote for the this sort of rational function is called a slant asympotote.

The equation for the slant asymptote is the polynomial part of the rational that we get after doing the long division.

y = (- x ² + 3x + 9)/(x - 1)

y  = (- x + 2) + 11/(x-1)

y  = -x + 2 is slant asympotote.

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

x	y = (- x 2 + 3x + 9)/(x - 1)	(x , y )
-1	y = [-1 2 +3(1)+9]/(1-1 ) = 1/0	not defined
2	y = [-2 2 +3(2)+9]/(2-1 ) = 11	(2, 11)
3	y = [-3 2 +3(3)+9]/(3-1 ) = 9/2 = 4.5	(3,4.5)

Graph

1) Draw the coordinate plane.

2) Next dash the slant and vertical asympototes.

3) Plot the x , y intercepts 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 of the function is all real numbers.