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
Transcript text: (b) What is the domain of the given function?
The domain is $(-\infty, 0) \cup(0, \infty)$.
(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 $\square$
Solution
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 (−∞,0)∪(0,∞), we should consider the function's behavior as it approaches zero from both sides and as it extends towards positive and negative infinity.
Step 1: Define the Function and Domain
We are given a function f(x)=x1 with the domain (−∞,0)∪(0,∞). This means the function is not defined at x=0.
Step 2: Analyze the Limits
To determine the range, we need to analyze the behavior of the function as x approaches critical points:
As x→0+, f(x)→∞
As x→0−, f(x)→−∞
As x→∞, f(x)→0
As x→−∞, f(x)→0
Step 3: Determine the Range
From the limits, we observe:
The function approaches ∞ as x approaches 0 from the positive side.
The function approaches −∞ as x approaches 0 from the negative side.
The function approaches 0 as x approaches ∞ or −∞.
Thus, the function f(x)=x1 can take any value except 0.