Consider the rational function...

Question

Consider the rational function f(x) = x²-2x-3 / x³-3x² .

 

a) Find all vertical and horizontal asymptotes of this rational function.

 

b) Find all x-intercepts and y-intercepts of the graph of this rational function.

 

c) State the domain of this rational function.

 

d) Does the graph of this rational function have a hole?  If so, what are the x and y-coordinates of the hole?

 

e) Determine when the graph of this rational function will be above and below the x-axis.

 

f) Sketch the graph of this rational function, being sure to draw all the features found above.

Answer 1

f(x) = (x + 1)(x - 3) / ((x² (x - 3))

 

One zero at (-1, 0)

Hole at x = 3

 

D = ℜ - {0, 3}

 

For x ≠ 3, f(x) = (x + 1) / x²

Vertical asymptote at x = 0.

 

For x > 0, horizontal asymptote at y = 0.

For x < -1, horizontal asymptote at y = 0, from below.

 

Minimum (-2, -1/4)