How do you identify all asymptotes or holes for #f(x)=1/(-2x-2)#?

Question

Lily Anderson · Accepted Answer

#"vertical asymptote at " x=-1#
#"horizontal asymptote at " y=0# The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote. solve: #-2x-2=0rArrx=-1" is the asymptote"# Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" ( a constant)"# divide terms on numerator/denominator by x #f(x)=(1/x)/((2x)/x-2/x)=(1/x)/(2-2/x)# as #xto+-oo,f(x)to0/(2-0)# #rArry=0" is the asymptote"# Holes occur when there are duplicate factors on the numerator/denominator. This is not the case here hence there are no holes. graph{1/(-2x-2) [-10, 10, -5, 5]}

Lily Anderson · Answer

undefined To identify all asymptotes or holes for $ f(x) = \frac{1}{-2x - 2} $, we first need to determine the behavior of the function as $ x $ approaches certain values.

1. Vertical Asymptote:
   - Set the denominator equal to zero and solve for $ x $. This is because vertical asymptotes occur where the function approaches positive or negative infinity as $ x $ approaches a certain value.
   - For $ f(x) = \frac{1}{-2x - 2} $, the denominator $ -2x - 2 $ equals zero when $ x = -1 $.
   - Thus, the vertical asymptote is at $ x = -1 $.

2. Horizontal Asymptote:
   - To find the horizontal asymptote, examine the behavior of the function as $ x $ approaches positive or negative infinity.
   - For rational functions like this one, the degree of the numerator and the degree of the denominator determine the horizontal asymptote.
   - Since the degree of the numerator is 0 and the degree of the denominator is 1, the horizontal asymptote is at $ y = 0 $ (the $ x $-axis).

3. Hole:
   - To identify a hole, check if there are any common factors in the numerator and the denominator that can be canceled out.
   - In this case, there are no common factors that can be canceled out, so there is no hole in the graph.

In summary:
- Vertical asymptote: $ x = -1 $
- Horizontal asymptote: $ y = 0 $
- No hole in the graph.