Question

What is the domain of the function (using interval notation is recommended)?

Answer 1

The domain of a function represents all possible input values that the function can accept without encountering mathematical issues. To provide an accurate domain, it's necessary to consider any restrictions or limitations imposed on the function's operations or expressions.

Step-by-step explanation:

The domain of a function is determined by examining any restrictions or limitations on the input values that would result in undefined or impossible operations within the function. Common restrictions include division by zero, square roots of negative numbers (for real numbers), or expressions that result in non-real values within the function.

For example, in rational functions, the denominator cannot be zero as it would lead to division by zero, causing the function to be undefined. Also, square roots of negative numbers are not real numbers within the domain of real functions.

In trigonometric or logarithmic functions, restrictions may arise from the nature of these functions. For instance, the domain of a trigonometric function might be limited to a specific range of angles or real numbers.

Understanding the nature of the function and the mathematical operations involved is crucial in determining its domain accurately.

Answer 2

The domain of the function is (-∞, ∞).

Step-by-step explanation:

In mathematical terms, the domain of a function refers to the set of all possible input values for which the function is defined. In this case, when we express the domain in interval notation as (-∞, ∞), it indicates that the function is defined for all real numbers. The symbol "∞" represents infinity, and the open intervals (-∞, ∞) imply that the function is valid for any real number, including negative and positive infinity.

To further clarify, an open interval does not include the endpoints, so (-∞, ∞) implies that there are no restrictions on the input values; the function is continuous and defined for any real number you choose. This expansive domain is common for functions like linear functions, polynomials, and many transcendental functions, where there are no singular points or values that cause the function to be undefined. Therefore, the interval notation succinctly captures the comprehensive nature of the function's domain, encompassing the entire real number line.