The problem asks to find the domain and range of the function $f(x) = \frac{1}{x}$.

1. Problem Description

The problem asks to find the domain and range of the function $f(x) = \frac{1}{x}$.

2. Solution Steps

Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function $f(x) = \frac{1}{x}$, the function is defined for all real numbers except when the denominator is equal to zero. Since we cannot divide by zero, $x$ cannot be equal to

0. Therefore, the domain is all real numbers except

0.

Range: The range of a function is the set of all possible output values (y-values) that the function can produce.

Let $y = \frac{1}{x}$. Then, $x = \frac{1}{y}$. Since $x$ can be any real number except 0, $\frac{1}{y}$ can be any real number except

0. This implies that $y$ can be any real number except

0. Thus, the range is all real numbers except

0. We can also reason that as $x$ approaches infinity, $f(x)$ approaches

0. Similarly, as $x$ approaches negative infinity, $f(x)$ approaches

0. Also, as $x$ approaches 0 from the positive side, $f(x)$ approaches infinity. And as $x$ approaches 0 from the negative side, $f(x)$ approaches negative infinity. Since the function is continuous for all non-zero $x$, $f(x)$ takes on all real values except

0.

3. Final Answer

Domain: All real numbers except

0. Range: All real numbers except 0.