Let's analyze the rational function \( f(x) = \frac{x}{x^2} \).
Domain:
The domain of a function consists of all the input values (x-values) for which the function is defined. In this case, \( f(x) = \frac{x}{x^2} \) is undefined when the denominator is zero.
Set the denominator equal to zero: \[ x^2 = 0 \]
Solving for \( x \), we find: \[ x = 0 \]
This means \( f(x) \) is undefined at \( x = 0 \). Therefore, the domain of the function is all real numbers except zero. In interval notation, this can be expressed as: \[ (-\infty, 0) \cup (0, \infty) \]
Range:
To find the range, we can rewrite the function: \[ f(x) = \frac{x}{x^2} = \frac{1}{x} \]
This function, \( f(x) = \frac{1}{x} \), is defined for all \( x \neq 0 \). The output values (y-values) for this function can be determined as:
- As \( x \) approaches zero from either the positive or negative side, \( f(x) \) approaches infinity or negative infinity, respectively.
- The function can take any real number except zero.
Therefore, the range of the function is: \[ (-\infty, 0) \cup (0, \infty) \]
Final Answers:
- The domain is \( (-\infty, 0) \cup (0, \infty) \).
- The range is \( (-\infty, 0) \cup (0, \infty) \).