Question

activity
in this activity, you will analyze transformations of the parent rational function, represented by an equation or a ta
question 1
consider the rational function $f(x) = \frac{1}{x + 1}$.
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 $x = -1$.

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

Step4: Determine range

Function never equals 0: $(-\infty, 0) \cup (0, \infty)$

Step5: Check for oblique asymptote

Degree of numerator < denominator: no oblique asymptote.

Step6: Check for holes

No common factors in numerator/denominator: no hole.

Answer:

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