How do you graph #f(x)= (x^2-100)/(x+10)#?

Question

Abigail Allen · Accepted Answer

#f(x) = (x^2-100)/(x+10) = ((x-10)(x+10))/(x+10) = x-10#

with excluded value #x=-10#.

This is a straight line of slope #1# passing through #(0, -10)# and #(10, 0)# with excluded point #(-10, -20)# #f(x) = (x^2-100)/(x+10)# #= ((x-10)(x+10))/(x+10)# #= x-10# with exclusion #x!=-10#. The graph of #f(x)# is like the graph of #x-10# except that #f(x)# is not defined at the point #(-10, -20)# All the limits as you approach that point behave well, it's just that #f(-10)# is undefined as it's equal to #0/0#. graph{(x^2-100)/(x+10) [-37.5, 42.5, -24, 16]}

Ethan Adkins · Answer

undefined To graph the function f(x) = (x^2-100)/(x+10), follow these steps:

1. Determine the domain of the function by finding the values of x for which the denominator (x+10) is equal to zero. In this case, x cannot be -10.

2. Identify any vertical asymptotes by examining the behavior of the function as x approaches the values in the domain. In this case, there is a vertical asymptote at x = -10.

3. Find the x-intercepts by setting the numerator (x^2-100) equal to zero and solving for x. In this case, the x-intercepts are x = -10 and x = 10.

4. Determine the y-intercept by evaluating the function at x = 0. In this case, the y-intercept is y = -10.

5. Plot the vertical asymptote, x-intercepts, and y-intercept on the coordinate plane.

6. Analyze the behavior of the function as x approaches positive and negative infinity. In this case, as x approaches positive or negative infinity, the function approaches the horizontal line y = x.

7. Sketch the graph by connecting the plotted points and considering the behavior of the function.

The graph of f(x) = (x^2-100)/(x+10) will have a vertical asymptote at x = -10, x-intercepts at x = -10 and x = 10, a y-intercept at y = -10, and will approach the line y = x as x approaches positive or negative infinity.