How do you graph and find the discontinuities of #1 /(x+6)#?

Question

Julia Adams · Accepted Answer

graph{1/(x+6) [-10, 10, -5, 5]} and #x=-6# Essentially it's transforming the graph #y=1/x# into #y=1/(x+6)# by translating #x# #6# units in the negative #x#-direction. Discontinuities occur when the function is undefined, i.e. #x=-6#, at which the graph is undefined. You can show this by drawing the vertical like #x=6# and showing that this line never intersects the graph of #y=1/(x+6)#

Eva Abramson · Answer

undefined To graph and find the discontinuities of the function 1/(x+6), follow these steps:

1. Start by determining the vertical asymptote(s) of the function. Set the denominator (x+6) equal to zero and solve for x. In this case, x = -6 is the vertical asymptote.

2. Next, find the horizontal asymptote of the function. Since the degree of the numerator (which is 1) is less than the degree of the denominator (which is 1), the horizontal asymptote is y = 0.

3. Plot the vertical asymptote at x = -6 and the horizontal asymptote at y = 0 on the coordinate plane.

4. To graph the function, choose some x-values on both sides of the vertical asymptote (-6) and evaluate the corresponding y-values using the function. Plot these points on the graph.

5. Connect the plotted points smoothly, approaching the vertical asymptote without crossing it.

6. The graph of 1/(x+6) will have a hole at x = -6 since the function is undefined at that point. Indicate this by leaving an open circle at the corresponding point on the graph.

7. The graph will approach the horizontal asymptote (y = 0) as x approaches positive or negative infinity.

This completes the process of graphing and identifying the discontinuity of the function 1/(x+6).