Question

The verbal description represents a rational function.

f of x equals x divided by x squared

What are the domain and range of the rational function?

Select the correct answers from the lists.

(1 point)
The domain is
.

The range is
.

Answer 1

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 \( (-\infty, 0) \cup (0, \infty) \).