Question

the domain of the function is
the range of the function is
$f(x) = \frac{1}{x}$

Explanation:

Step1: Analyze the domain of \( f(x)=\frac{1}{x} \)

The function \( f(x)=\frac{1}{x} \) is undefined when the denominator \( x = 0 \). From the graph, we can see the hyperbola does not touch the \( y \)-axis (where \( x = 0 \)). So the domain is all real numbers except \( 0 \), which can be written as \( (-\infty, 0) \cup (0, \infty) \).

Step2: Analyze the range of \( f(x)=\frac{1}{x} \)

For \( y=\frac{1}{x} \), we can rearrange to \( x=\frac{1}{y} \). Here, \( y \) cannot be \( 0 \) (since that would make \( x \) undefined). From the graph, the hyperbola does not touch the \( x \)-axis (where \( y = 0 \)). So the range is all real numbers except \( 0 \), which is \( (-\infty, 0) \cup (0, \infty) \).

Answer:

The domain of the function is \( (-\infty, 0) \cup (0, \infty) \)
The range of the function is \( (-\infty, 0) \cup (0, \infty) \)