follow the steps for graphing a rational function to graph the function ( r(x)=\frac{3x + 3}{7x+35} ) (type…

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{3x + 3}{7x+35} ) (type an equation. use integers or fractions for any numbers in the equation.) o c. the function has three vertical asymptotes. the leftmost asymptote is, the middle asymptote is, and the rightmost asymptote is (type equations. use integers or fractions for any numbers in the equations.) o d. the function has no vertical asymptote. determine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice. o a. there is a hole in the graph at the point (type an ordered pair using integers or fractions.) o b. there are no holes in the graph determine the behavior of the graph on either side of any vertical asymptotes, if one exists. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice o a. 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= ) (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) o b. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x= ) (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) o c. it approaches ( infty ) on one side of the asymptote(s) at ( x=-5 ) and ( -infty ) on the other. (type an integer or a simplified fraction. use a comma to separate answers as needed type each answer only once.) o d. there is no vertical asymptote determine the horizontal asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. o a. the function has one horizontal asymptote. ( \frac{3}{7} ) (type an equation. use integers or fractions for any numbers in the equation.) o b. the function has two horizontal asymptotes. the top asymptote is, and the bottom asymptote is (type equations. use integers or fractions for any numbers in the equations.) o c. the function has no horizontal asymptote.

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{3x + 3}{7x+35} ) (type an equation. use integers or fractions for any numbers in the equation.) o c. the function has three vertical asymptotes. the leftmost asymptote is, the middle asymptote is, and the rightmost asymptote is (type equations. use integers or fractions for any numbers in the equations.) o d. the function has no vertical asymptote. determine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice. o a. there is a hole in the graph at the point (type an ordered pair using integers or fractions.) o b. there are no holes in the graph determine the behavior of the graph on either side of any vertical asymptotes, if one exists. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice o a. 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= ) (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) o b. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x= ) (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) o c. it approaches ( infty ) on one side of the asymptote(s) at ( x=-5 ) and ( -infty ) on the other. (type an integer or a simplified fraction. use a comma to separate answers as needed type each answer only once.) o d. there is no vertical asymptote determine the horizontal asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. o a. the function has one horizontal asymptote. ( \frac{3}{7} ) (type an equation. use integers or fractions for any numbers in the equation.) o b. the function has two horizontal asymptotes. the top asymptote is, and the bottom asymptote is (type equations. use integers or fractions for any numbers in the equations.) o c. the function has no horizontal asymptote.

Answer

Explanation:

Step1: Find vertical asymptote

For a rational function (R(x)=\frac{f(x)}{g(x)}), vertical asymptote occurs when (g(x) = 0). Set (7x+35=0), then (7x=-35), (x = - 5).

Step2: Check for holes

Factor numerator and denominator. (R(x)=\frac{3x + 3}{7x+35}=\frac{3(x + 1)}{7(x + 5)}). There is no common factor (other than 1) in numerator and denominator, so no holes.

Step3: Analyze behavior near vertical asymptote

As (x\to - 5^{-}) (approaching -5 from the left), (7x+35\to0^{-}) and (3x + 3\to3\times(-5)+3=-12). So (R(x)=\frac{3x + 3}{7x+35}\to+\infty). As (x\to - 5^{+}) (approaching -5 from the right), (7x+35\to0^{+}) and (3x + 3\to3\times(-5)+3=-12). So (R(x)=\frac{3x + 3}{7x+35}\to-\infty).

Step4: Find horizontal asymptote

For (R(x)=\frac{3x+3}{7x + 35}), since the degree of numerator (n = 1) and degree of denominator (m=1). The horizontal asymptote is (y=\frac{a_{n}}{b_{m}}), where (a_{n}) is the leading coefficient of numerator ((a_{n}=3)) and (b_{m}) is the leading coefficient of denominator ((b_{m}=7)). So (y=\frac{3}{7})

Answer:

Vertical asymptote: (x=-5); Holes: No holes (Option B); Behavior near vertical asymptote: It approaches (\infty) on one side of the asymptote at (x = - 5) and (-\infty) on the other (Option C); Horizontal asymptote: (y=\frac{3}{7}) (Option A)