Question

the function ( f(x)=\frac{3}{x + 1} ) is a rational function

a find all vertical asymptotes
b create a rough sketch of the graph near each asymptote

a select the correct choice and. if necessary, fill in the answer box(es) to complete your choice.

( \bigcirc ) a. the function has one vertical asymptote,
(type an equation use integers or fractions for any numbers in the equation )

( \bigcirc ) b. the function has two vertical asymptotes the leftmost asymptote is and the rightmost asymptote is
(type equations use integers or fractions for any numbers in the equation )

( \bigcirc ) c. the function has three vertical asymptotes the leftmost asymptote is the middle asymptote is and the rightmost asymptote is
(type an equations use integers or fractions for any numbers in the equations )

( \bigcirc ) d. the function has no vertical asymptotes

Explanation:

Step1: Recall the definition of vertical asymptote

For a rational function \(y = \frac{f(x)}{g(x)}\), vertical asymptotes occur at the values of \(x\) that make the denominator \(g(x)=0\) (provided the numerator \(f(x)
eq0\) at those values).
For the function \(f(x)=\frac{3}{x + 1}\), set the denominator equal to zero: \(x+1=0\).

Step2: Solve the equation for \(x\)

Subtract \(1\) from both sides of the equation \(x + 1=0\). We get \(x=-1\).
Check that the numerator \(f(-1)=3
eq0\).

Answer:

A. The function has one vertical asymptote, \(x=-1\)