homework assignment #6\ncurrent score:\n20/25 points\ncategory: quiz\nnext\ncurrent learning objective…

homework assignment #6\ncurrent score:\n20/25 points\ncategory: quiz\nnext\ncurrent learning objective: identifying vertical asymptotes of rational functions\nquestion 24\npractice similar questions\nscore: 0 of 1 point\ndetermine the vertical asymptotes and holes (removable points of discontinuity) of the rational function shown below.\n f(x)=\frac{x - 6}{(x + 10)(x - 9)} \nif there is not a hole or asymptote, record dne as your answer.\nholes\n x=\nvertical asymptotes\n(enter the asymptote with the smaller value first)\n x=\n x=

homework assignment #6\ncurrent score:\n20/25 points\ncategory: quiz\nnext\ncurrent learning objective: identifying vertical asymptotes of rational functions\nquestion 24\npractice similar questions\nscore: 0 of 1 point\ndetermine the vertical asymptotes and holes (removable points of discontinuity) of the rational function shown below.\n f(x)=\frac{x - 6}{(x + 10)(x - 9)} \nif there is not a hole or asymptote, record dne as your answer.\nholes\n x=\nvertical asymptotes\n(enter the asymptote with the smaller value first)\n x=\n x=

Answer

Explanation:

Step1: Find the domain restrictions

Set the denominator ((x + 10)(x - 9)=0). Using the zero - product property (a\times b = 0) implies (a = 0) or (b = 0). So (x+10 = 0) gives (x=-10) and (x - 9=0) gives (x = 9).

Step2: Check for holes

A hole occurs when a factor in the numerator and a factor in the denominator cancel out. The numerator is (x - 6). Since there is no common factor between (x - 6) and ((x + 10)(x - 9)), there is no hole.

Step3: Determine vertical asymptotes

Vertical asymptotes occur at the values of (x) that make the denominator zero (when there is no common factor with the numerator). Since (x=-10) and (x = 9) make the denominator zero and there are no common factors with the numerator, these are the vertical asymptotes.

Answer:

Holes: (x=\text{DNE}) Vertical Asymptotes: (x=-10), (x = 9)