follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x + 3}{x(x + 7)}…

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x + 3}{x(x + 7)} ).\ndetermine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.\na. there is a hole in the graph at the point (square).\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.\na. it approaches (infty) on one side of the asymptote(s) at ( x=square ) and (-infty) on the other. it approaches either (infty) or (-infty) on both sides of the asymptote(s) at ( x=square ).\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)\nb. it approaches (infty) on one side of the asymptote(s) at ( x=square ) and (-infty) on the other.\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)\nc. it approaches either (infty) or (-infty) on both sides of the asymptote(s) at ( x=square ).\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)\nd. 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, (square).\n(type an equation. use integers or fractions for any numbers in the equation.)\nb. the function has two horizontal asymptotes. the top asymptote is (square), and the bottom asymptote is (square).

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x + 3}{x(x + 7)} ).\ndetermine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.\na. there is a hole in the graph at the point (square).\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.\na. it approaches (infty) on one side of the asymptote(s) at ( x=square ) and (-infty) on the other. it approaches either (infty) or (-infty) on both sides of the asymptote(s) at ( x=square ).\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)\nb. it approaches (infty) on one side of the asymptote(s) at ( x=square ) and (-infty) on the other.\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)\nc. it approaches either (infty) or (-infty) on both sides of the asymptote(s) at ( x=square ).\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)\nd. 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, (square).\n(type an equation. use integers or fractions for any numbers in the equation.)\nb. the function has two horizontal asymptotes. the top asymptote is (square), and the bottom asymptote is (square).

Answer

Explanation:

Step1: Check for holes

A hole occurs when a factor cancels out in the numerator and denominator. The function ( R(x)=\frac{x + 3}{x(x + 7)} ) has no common factors in the numerator and denominator. So, there are no holes.

Step2: Check for vertical asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero). Set ( x(x + 7)=0 ), which gives ( x = 0 ) and ( x=-7 ).

Step3: Check for horizontal asymptotes

For a rational function ( \frac{f(x)}{g(x)} ) where ( f(x)=x + 3) (degree ( n = 1 )) and ( g(x)=x^{2}+7x) (degree ( m = 2 )). Since ( n<m ), the horizontal asymptote is ( y = 0 ).

Answer:

  • Hole: B. There are no holes in the graph.
  • Vertical asymptotes: B. The function has vertical asymptotes ( x = 0 ) and ( x=-7 ).
  • Horizontal asymptotes: A. The function has one horizontal asymptote ( y = 0 ).