Questions: What is the domain of the given function? The domain is (-∞, 0) ∪(0, ∞). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) What is the range of the given function? The range is

Solution Steps

To determine the range of the given function, we need to analyze the behavior of the function over its domain. Since the domain is $(-\infty, 0) \cup (0, \infty)$, we should consider the function's behavior as it approaches zero from both sides and as it extends towards positive and negative infinity.

We are given a function \( f(x) = \frac{1}{x} \) with the domain \( (-\infty, 0) \cup (0, \infty) \). This means the function is not defined at \( x = 0 \).

To determine the range, we need to analyze the behavior of the function as \( x \) approaches critical points:

• As \( x \to 0^+ \), \( f(x) \to \infty \)
• As \( x \to 0^- \), \( f(x) \to -\infty \)
• As \( x \to \infty \), \( f(x) \to 0 \)
• As \( x \to -\infty \), \( f(x) \to 0 \)

From the limits, we observe:

• The function approaches \( \infty \) as \( x \) approaches \( 0 \) from the positive side.
• The function approaches \( -\infty \) as \( x \) approaches \( 0 \) from the negative side.
• The function approaches \( 0 \) as \( x \) approaches \( \infty \) or \( -\infty \).

Thus, the function \( f(x) = \frac{1}{x} \) can take any value except \( 0 \).

Final Answer