Practice!

Sample Problem

g(x)=ln(1-x)

What is the domain of g?

all real numbers

(0, ∞)

(1, ∞)

(-∞, 1)

What is the x-intercept of g?

(0, 0)

(0, 1)

(0, -1)

no x-intercept

Where are the vertical asymptotes of g?

x=-1

x=0

x=1

no vertical asymptote

Solution

Because ln(1-x) is defined only if 1-x>0, this means that x<1 for the function to be defined. The domain is (-∞, 1).

The x-intercept of g is when g(x)=ln(1-x)=0. Converted into exponential form, this is e0=1-x.

e0=1=1-x

→x=0. The x-intercept of g is at (0, 0)

The vertical asymptote of g occurs when the inside of the parentheses in the ln function hits 0. In this case, our function is g(x)=ln(1-x).

1-x=0 → at x=1, there is a vertical asymptote because the domain is restricted x<1, and as x approaches 1 from the left, f(x) approaches -∞.