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 done checkmark

Explanation:

Step1: Find vertical asymptote

A vertical asymptote occurs where the function is undefined. For $f(x)=\frac{c}{x}$, the denominator is 0 when $x=0$, so this is the vertical asymptote.

Step2: Find horizontal asymptote

A horizontal asymptote is found by taking the limit as $x\to\pm\infty$. $\lim_{x\to\pm\infty}\frac{c}{x}=0$, so this is the horizontal asymptote.

Step3: Find the domain

The domain is all real numbers except where the function is undefined. Since $x=0$ makes the denominator 0, the domain is all real numbers except 0.

Step4: Find the range

The range is all possible output values of $f(x)$. For any non-zero real number $y$, we can solve $y=\frac{c}{x}$ to get $x=\frac{c}{y}$, which is defined for all non-zero $y$. So the range is all real numbers except 0.

Answer:

The vertical asymptote: $x=0$
The horizontal asymptote: $y=0$
The domain: All real numbers except $x=0$ (or $\mathbb{R}\setminus\{0\}$)
The range: All real numbers except $y=0$ (or $\mathbb{R}\setminus\{0\}$)