Question

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

Explanation:

Step1: Find vertical asymptote

For a rational function \(y = \frac{c}{x}\), the denominator cannot be zero. When \(x = 0\), the function is undefined. So, the vertical asymptote is \(x = 0\).

Step2: Find horizontal asymptote

We use the limit as \(x\to\pm\infty\). \(\lim_{x\to\pm\infty}\frac{c}{x}=0\). So, the horizontal asymptote is \(y = 0\).

Step3: Find domain

The function \(y=\frac{c}{x}\) is defined for all real numbers except \(x = 0\). So, the domain is \(\{x|x
eq0,x\in R\}\).

Step4: Find range

Let \(y=\frac{c}{x}\), then \(x=\frac{c}{y}\). \(y
eq0\) (because if \(y = 0\), then \(\frac{c}{y}\) is undefined). So, the range is \(\{y|y
eq0,y\in R\}\).

Answer:

• The vertical asymptote: \(x = 0\)
• The horizontal asymptote: \(y = 0\)
• The domain: \(\{x|x

eq0,x\in R\}\)

• The range: \(\{y|y

eq0,y\in R\}\)