Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x^2? What is its range?

Verified step by step guidance

1. <Step 1: Understand the function. The function given is \( f(x) = \frac{1}{x^2} \).>
2. <Step 2: Determine the domain. The domain of a function is the set of all possible input values (x-values) that will not make the function undefined.>
3. <Step 3: Identify restrictions for the domain. Since the function has \( x^2 \) in the denominator, it cannot be zero. Therefore, \( x \neq 0 \).>
4. <Step 4: State the domain. The domain of \( f(x) = \frac{1}{x^2} \) is all real numbers except zero, which can be written as \( (-\infty, 0) \cup (0, \infty) \).>
5. <Step 5: Determine the range. Since \( x^2 \) is always positive for any real number \( x \neq 0 \), \( \frac{1}{x^2} \) is always positive. Thus, the range is \( (0, \infty) \).>

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function ƒ(x) = 1/x², the domain excludes any values that make the denominator zero, which in this case is x = 0. Therefore, the domain is all real numbers except zero, expressed as (-∞, 0) ∪ (0, ∞).

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For ƒ(x) = 1/x², as x approaches zero from either side, the function approaches infinity, and as x moves away from zero, the function approaches zero but never reaches it. Thus, the range is all positive real numbers, expressed as (0, ∞).

Vertical Asymptote

A vertical asymptote is a line that a graph approaches but never touches or crosses, indicating where a function's value becomes unbounded. In the case of ƒ(x) = 1/x², there is a vertical asymptote at x = 0, which signifies that as x approaches zero, the function's value increases without bound, reinforcing the understanding of the domain and range.