How do you graph f(x)=2/(x-1)f(x)=2x1 using holes, vertical and horizontal asymptotes, x and y intercepts?

1 Answer
Oct 14, 2017

No holes.
Vertical asymptote: x=1x=1
Horizontal asymptote: y=0y=0
No xx intercepts.
yy-intercept: -22

Explanation:

Denote f(x)f(x) as (n(x))/(d(x)n(x)d(x)

There are no holes since there are no common factors.

To find the vertical asymptote,
Solve d(x)=0d(x)=0
rArrx-1=0x1=0
x=1x=1

Therefore the vertical asymptote is x=1x=1.

To find the horizontal asymptote,
Compare the leading degree of n(x)n(x) and d(x)d(x).

For n(x)n(x), the degree is 00, because x^0*2x02 gives 22. Denote this as color(turquoise)nn
For d(x)d(x), the degree is 11 (since x^1x1). Denote this as color(magenta)mm

When n < mn<m, the xx-axis (that is, y=0y=0) is the horizontal asymptote.

To find the xx intercept, plug in 00 for yy and solve for xx.
rArr0=2/(x-1)0=2x1
There are no xx intercepts.

To find the yy intercept, plug in 00 of xx and solve for yy.
rArrf(x)=2/(0-1)f(x)=201
f(x)=-2f(x)=2
The yy-intercept is -22.

graph{2/(x-1 [-100, 100, -5, 5]}