Question

consider the function $f(x) = \frac{c}{x}$, where $c$ is a nonzero real number. the vertical asymptote dropdown. the horizontal asymptote dropdown. the domain dropdown. the range dropdown.

Explanation:

Step1: Find Vertical Asymptote

A vertical asymptote occurs where the function is undefined (denominator is zero). For \( f(x)=\frac{c}{x} \), denominator \( x = 0 \) makes it undefined. So vertical asymptote is \( x = 0 \).

Step2: Find Horizontal Asymptote

For a rational function \( \frac{N(x)}{D(x)} \), if degree of \( N(x) \) < degree of \( D(x) \), horizontal asymptote is \( y = 0 \). Here \( N(x)=c \) (degree 0), \( D(x)=x \) (degree 1), so horizontal asymptote is \( y = 0 \).

Step3: Find Domain

Domain is all real numbers except where function is undefined. Since \( x = 0 \) is undefined, domain is \( \{x|x
eq0, x\in\mathbb{R}\} \) or \( (-\infty, 0)\cup(0, \infty) \).

Step4: Find Range

Range is all possible \( y \)-values. For \( y=\frac{c}{x} \), solving for \( x \) gives \( x=\frac{c}{y} \), so \( y \) can't be 0 (as \( x \) would be undefined). Thus range is \( \{y|y
eq0, y\in\mathbb{R}\} \) or \( (-\infty, 0)\cup(0, \infty) \).

Answer:

• Vertical Asymptote: \( x = 0 \)
• Horizontal Asymptote: \( y = 0 \)
• Domain: All real numbers except \( 0 \) (or \( (-\infty, 0)\cup(0, \infty) \))
• Range: All real numbers except \( 0 \) (or \( (-\infty, 0)\cup(0, \infty) \))