Rational Functions...2

[nycmathguy]

Section 2.6
Question 28

In Exercises 17–38, (a) state
the domain of the function, (b) identify all
intercepts, (c) find any vertical or horizontal
asymptotes, and (d) plot additional solution
points as needed to sketch the graph of the rational function.

Note: Part (d) will NOT be done by hand.

Part (a)

Domain = All real numbers except for x = -3 and x = 1.

Part (b)

Note: g(x) = y.

If x = 0, the y-intercept is the line y = 0 or g(x) = 0, which is the x-axis.

Let g(x) = 0.

0 = 3x/(x^2 + 2x - 3)

(x^2 + 2x - 3)(0) = 3x/(x^2 + 2x - 3)(x^2 + 2x - 3)/1

0 = 3x

0/3 = x

0 = x

The x-intercept is the line x = 0 aka the y-axis.

Part (c)

Top degree < bottom degree.
So, the horizontal asymptote is the line y = 0.

Note: The horizontal asymptote is also the y-intercept. Is this possible?

The vertical asymptotes are x = -3 and x = 1.

Part (d)

 

[MathLover1]

perfect

 

[nycmathguy]

perfect

Wish I had time to completely work out 25 through 36 but working 40 overnight hours takes it toll.