Provide a short answer to each question. What is the domain of the function $\text{ƒ}\left(x\right)=\frac{1}{x^2}$? What is its range?

Verified step by step guidance

1. Identify the domain by determining all values of \(x\) for which the function \(f(x) = \frac{1}{x^2}\) is defined. Since division by zero is undefined, exclude values that make the denominator zero.
2. Set the denominator equal to zero and solve: \(x^2 = 0\). This gives \(x = 0\), which must be excluded from the domain.
3. Conclude that the domain is all real numbers except \(x = 0\), which can be written as \((-\infty, 0) \cup (0, \infty)\).
4. To find the range, analyze the output values of \(f(x) = \frac{1}{x^2}\). Since \(x^2\) is always positive for \(x \neq 0\), \(\frac{1}{x^2}\) is also always positive.
5. Note that as \(x\) approaches zero, \(f(x)\) grows without bound, and as \(x\) approaches infinity or negative infinity, \(f(x)\) approaches zero but never reaches it. Therefore, the range is \((0, \infty)\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like ƒ(x) = 1/x², the domain excludes values that make the denominator zero, as division by zero is undefined.

Range of a Function

The range of a function is the set of all possible output values (y-values) the function can produce. For ƒ(x) = 1/x², since x² is always positive except at zero, the function outputs positive values, and the range reflects these possible outputs.

Behavior of Rational Functions with Squared Denominators

In functions like ƒ(x) = 1/x², the denominator is squared, making it always positive except at zero. This affects the function's behavior by ensuring outputs are positive and the function approaches infinity near zero, influencing both domain restrictions and range values.