Question

question 1
consider the rational function $f(x)=\frac{3}{x + 1}$.
part a
select all the statements that describe a key feature of function $f$.

• there is a horizontal asymptote at $y=-1$.
• the domain is $(-\infty,0)\cup(0,\infty)$.
• there is a vertical asymptote at $x=-1$.
• there is an oblique asymptote.
• the range is $(-\infty,0)\cup(0,\infty)$.
• there is a hole in the graph at $x=-1$.
• there is a vertical asymptote at $x = 0$.

Explanation:

Step1: Find vertical asymptote

Set denominator to 0: $x+1=0 \implies x=-1$

Step2: Find horizontal asymptote

Compare degrees: numerator degree 0, denominator degree 1. Horizontal asymptote is $y=0$.

Step3: Determine domain

Exclude $x=-1$: Domain is $(-\infty, -1) \cup (-1, \infty)$

Step4: Determine range

The function can never equal 0, so range is $(-\infty, 0) \cup (0, \infty)$

Step5: Check for oblique asymptote

Oblique asymptotes exist when numerator degree = denominator degree +1; not true here.

Step6: Check for holes

No common factors in numerator/denominator, so no holes.

Answer:

• There is a vertical asymptote at $x = -1$.
• The range is $(-\infty, 0) \cup (0, \infty)$.