find the vertical asymptotes and holes (removable discontinuities) of the rational function.\n\n…

find the vertical asymptotes and holes (removable discontinuities) of the rational function.\n\n f(x)=\frac{x^{2}-12 x + 32}{x^{2}-4 x - 32} \n\nthe holes are at ( x = )\n\nthe vertical asymptotes are ( x = )\n\nif there is more than one of each, enter your answers separated by a comma. if there is not a hole or asymptote, enter dne.\n\nquestion help:

find the vertical asymptotes and holes (removable discontinuities) of the rational function.\n\n f(x)=\frac{x^{2}-12 x + 32}{x^{2}-4 x - 32} \n\nthe holes are at ( x = )\n\nthe vertical asymptotes are ( x = )\n\nif there is more than one of each, enter your answers separated by a comma. if there is not a hole or asymptote, enter dne.\n\nquestion help:

Answer

Explanation:

Step1: Factor numerator and denominator

Factor (x^{2}-12x + 32=(x - 4)(x - 8)) Factor (x^{2}-4x-32=(x + 4)(x - 8)) So (f(x)=\frac{(x - 4)(x - 8)}{(x + 4)(x - 8)})

Step2: Simplify the function

Cancel out the common factor ((x - 8)) (for (x\neq8)). The simplified function is (f(x)=\frac{x - 4}{x + 4}), (x\neq8)

Step3: Find holes and vertical asymptotes

  • Holes: A hole occurs when a factor cancels out. The hole is at (x = 8)
  • Vertical asymptotes: Set the denominator of the simplified function (x+4 = 0), so (x=-4)

Answer:

The holes are at (x = 8). The vertical asymptotes are (x=-4)