Find the domain of each function. f(x) = 1/(x²+1) - 1/(x²-1)

Verified step by step guidance

1. Step 1: Recall that the domain of a function consists of all real numbers for which the function is defined. For rational functions, the function is undefined when the denominator equals zero. Identify the denominators in the given function: \( f(x) = \frac{1}{x^2+1} - \frac{1}{x^2-1} \).
2. Step 2: Analyze the first term \( \frac{1}{x^2+1} \). The denominator \( x^2+1 \) is never zero for any real value of \( x \), because \( x^2+1 \geq 1 \) for all real \( x \). Therefore, this term does not restrict the domain.
3. Step 3: Analyze the second term \( \frac{1}{x^2-1} \). The denominator \( x^2-1 \) equals zero when \( x^2 = 1 \). Solve \( x^2 = 1 \) to find \( x = \pm 1 \). These values make the denominator zero, so \( x = 1 \) and \( x = -1 \) must be excluded from the domain.
4. Step 4: Combine the results from Steps 2 and 3. The only restrictions on the domain come from the second term, where \( x = \pm 1 \) must be excluded. For all other real values of \( x \), the function is defined.
5. Step 5: Express the domain in interval notation, excluding \( x = \pm 1 \). The domain is \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).

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 rational functions, the domain is restricted by values that make the denominator zero, as division by zero is undefined. Understanding the domain is crucial for determining where the function can be evaluated without resulting in undefined expressions.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function f(x) = 1/(x^2+1) - 1/(x^2-1), each term is a rational function. Analyzing rational functions involves identifying points where the denominator is zero, which directly affects the domain and behavior of the function.

Finding Restrictions on the Domain

To find the domain of a function, one must identify any restrictions that arise from the function's structure. For the function f(x), we need to set the denominators of each term to zero and solve for x. The values that make the denominators zero are excluded from the domain, leading to a complete understanding of where the function is valid.