follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x + 4}{x(x + 10)}…

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x + 4}{x(x + 10)} ).\noc. the function has one vertical asymptote,\n(type an equation. use integers or fractions for any numbers in the equation)\nod. the function has no vertical asymptote\ndetermine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.\noa. there is a hole in the graph at the point\n(type an ordered pair using integers or fractions)\nb. there are no holes in the graph\ndetermine the behavior of the graph on either side of any vertical asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice\noa. it approaches ( infty ) on one side of the asymptote(s) at ( x= ) and ( -infty ) on the other. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x= )\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once)\nob. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x= )\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once)\nc. it approaches ( infty ) on one side of the asymptote(s) at ( x=0,-10 ) and ( -infty ) on the other\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once)\nod. the function has no vertical asymptote\ndetermine the horizontal asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice.\na. the function has one horizontal asymptote.\n(type an equation. use integers or fractions for any numbers in the equation)\nob. the function has two horizontal asymptotes. the top asymptote is, and the bottom asymptote is\n(type equations. use integers or fractions for any numbers in the equations)\noc. the function has no horizontal asymptote

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x + 4}{x(x + 10)} ).\noc. the function has one vertical asymptote,\n(type an equation. use integers or fractions for any numbers in the equation)\nod. the function has no vertical asymptote\ndetermine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.\noa. there is a hole in the graph at the point\n(type an ordered pair using integers or fractions)\nb. there are no holes in the graph\ndetermine the behavior of the graph on either side of any vertical asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice\noa. it approaches ( infty ) on one side of the asymptote(s) at ( x= ) and ( -infty ) on the other. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x= )\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once)\nob. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x= )\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once)\nc. it approaches ( infty ) on one side of the asymptote(s) at ( x=0,-10 ) and ( -infty ) on the other\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once)\nod. the function has no vertical asymptote\ndetermine the horizontal asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice.\na. the function has one horizontal asymptote.\n(type an equation. use integers or fractions for any numbers in the equation)\nob. the function has two horizontal asymptotes. the top asymptote is, and the bottom asymptote is\n(type equations. use integers or fractions for any numbers in the equations)\noc. the function has no horizontal asymptote

Answer

Explanation:

Step1: Find vertical asymptotes

Set the denominator (x(x + 10)=0). Solving (x=0) and (x=- 10) gives the vertical asymptotes.

Step2: Check for holes

Since the numerator (x + 4) and denominator (x(x + 10)) have no common factors (other than (1)), there are no holes.

Step3: Analyze behavior near vertical asymptotes

For (x = 0):

  • As (x\to0^{+}), (R(x)=\frac{x + 4}{x(x + 10)}\to+\infty) (because numerator (\to4) and denominator (\to0^{+}))
  • As (x\to0^{-}), (R(x)=\frac{x + 4}{x(x + 10)}\to-\infty) (because numerator (\to4) and denominator (\to0^{-}))

For (x=-10):

  • As (x\to - 10^{+}), (R(x)=\frac{x + 4}{x(x + 10)}\to-\infty) (numerator (\to - 6) and denominator (\to0^{-}))
  • As (x\to - 10^{-}), (R(x)=\frac{x + 4}{x(x + 10)}\to+\infty) (numerator (\to - 6) and denominator (\to0^{+}))

Step4: Find horizontal asymptote

The degree of the numerator (n = 1) and the degree of the denominator (m=2). When (n\lt m), the horizontal asymptote is (y = 0) (using the rule for rational functions (y=\frac{a_nx^n+\cdots}{b_mx^m+\cdots}), when (n\lt m), (y = 0))

Answer:

  • Vertical asymptotes: (x = 0) and (x=-10)
  • Holes: B. There are no holes in the graph
  • Behavior near vertical asymptotes:
    • At (x = 0): approaches (+\infty) on one side ((x\to0^{+})) and (-\infty) on the other ((x\to0^{-}))
    • At (x=-10): approaches (-\infty) on one side ((x\to - 10^{+})) and (+\infty) on the other ((x\to - 10^{-}))
  • Horizontal asymptote: A. The function has one horizontal asymptote (y = 0)