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

Question

Abigail Allen · Accepted Answer

No holes.
Vertical asymptote: #x=1#
Horizontal asymptote: #y=0#
No #x# intercepts.
#y#-intercept: #-2# Denote #f(x)# as #(n(x))/(d(x)# There are no holes since there are no common factors. To find the vertical asymptote, Solve #d(x)=0# #rArr##x-1=0# #x=1# Therefore the vertical asymptote is #x=1#. To find the horizontal asymptote, Compare the leading degree of #n(x)# and #d(x)#. For #n(x)#, the degree is #0#, because #x^0*2# gives #2#. Denote this as #color(turquoise)n# For #d(x)#, the degree is #1# (since #x^1#). Denote this as #color(magenta)m# When #n < m#, the #x#-axis (that is, #y=0#) is the horizontal asymptote. To find the #x# intercept, plug in #0# for #y# and solve for #x#. #rArr##0=2/(x-1)# There are no #x# intercepts. To find the #y# intercept, plug in #0# of #x# and solve for #y#. #rArr##f(x)=2/(0-1)# #f(x)=-2# The #y#-intercept is #-2#. graph{2/(x-1 [-100, 100, -5, 5]}

Avery Alexander · Answer

undefined To graph the function f(x) = 2/(x-1), we can start by identifying the vertical and horizontal asymptotes, holes, x-intercepts, and y-intercept.

Vertical asymptote:
The vertical asymptote occurs when the denominator of the function becomes zero. In this case, the denominator (x-1) becomes zero when x = 1. Therefore, the vertical asymptote is x = 1.

Horizontal asymptote:
To determine the horizontal asymptote, we need to compare the degrees of the numerator and denominator. In this case, the degree of the numerator is 0 (constant) and the degree of the denominator is 1. Since the degree of the denominator is greater, the horizontal asymptote is y = 0.

Hole:
A hole occurs when both the numerator and denominator have a common factor that can be canceled out. In this case, there is no common factor, so there is no hole.

X-intercept:
To find the x-intercept, we set y (or f(x)) equal to zero and solve for x. In this case, we have 2/(x-1) = 0. Since the numerator is never zero, there are no x-intercepts.

Y-intercept:
To find the y-intercept, we set x equal to zero and solve for y (or f(x)). In this case, we have f(0) = 2/(0-1) = -2. Therefore, the y-intercept is (0, -2).

To summarize:
- Vertical asymptote: x = 1
- Horizontal asymptote: y = 0
- Hole: None
- X-intercept: None
- Y-intercept: (0, -2)

Using this information, you can plot the graph of f(x) = 2/(x-1) accordingly.