Problem 86
Question
Find the domain of the function.$$f(x)=\sqrt[3]{16-x^{2}}$$.
Step-by-Step Solution
Verified Answer
The domain of the function \(f(x)=\sqrt[3]{16-x^{2}}\) is all real numbers, represented as (-∞ , ∞) in interval notation.
1Step 1: Understanding the Function
The given function is \(f(x)=\sqrt[3]{16-x^{2}}\), which is a cubic root function. A cubic root function can have any real numbers inside the cubic root, which means, the expression inside the cubic root \((16-x^{2})\) can be any real number. As there are no restrictions for the expression \(16 -x^{2}\), x can be any real number.
2Step 2: Finding the Domain of the Function
As established in Step 1, x can be any real number which means the domain of the function is all real numbers. In interval notation, this could be represented as (-∞ , ∞).
Key Concepts
Cubic Root FunctionInterval NotationReal Numbers
Cubic Root Function
A cubic root function, often represented as \(f(x) = \sqrt[3]{x}\), is a type of root function where any real number can be an operand inside the cubic root. Unlike square root functions, which only allow for non-negative operands to ensure a real result, cubic root functions accept both positive and negative inputs, as well as zero.
The function in the given exercise, \(f(x) = \sqrt[3]{16 - x^2}\), exhibits a key characteristic of cubic root functions—it can process any real number without causing undefined or imaginary results. This is because taking the cubic root of any real number will always result in a real number. Hence, whether the expression \(16 - x^2\) inside the cubic root is positive, negative, or zero, the outcome of the cubic root will be well-defined within the real number system.
The function in the given exercise, \(f(x) = \sqrt[3]{16 - x^2}\), exhibits a key characteristic of cubic root functions—it can process any real number without causing undefined or imaginary results. This is because taking the cubic root of any real number will always result in a real number. Hence, whether the expression \(16 - x^2\) inside the cubic root is positive, negative, or zero, the outcome of the cubic root will be well-defined within the real number system.
Interval Notation
Interval notation is a mathematical method used to describe a set of numbers along a continuum. It is frequently utilized to express domains and ranges of functions. The interval is denoted by a pair of numbers that are the end points of the interval, and can include a variety of brackets and parentheses depending on whether the endpoints are included in the set.
For example, in the provided exercise, the expression 'all real numbers' is equivalent to the interval notation (-∞, ∞), where the use of parentheses indicates that infinity is not a number that can be reached or included in the set, but rather a concept that extends beyond any finite boundary. Similarly, the notation \([a, b]\) would mean all numbers from 'a' to 'b', including 'a' and 'b'. In contrast, \((a, b]\) includes numbers from 'a' to 'b' but excludes 'a' while including 'b'. Understanding how to read and write in interval notation is essential for math students, especially when dealing with the domain and range of functions.
For example, in the provided exercise, the expression 'all real numbers' is equivalent to the interval notation (-∞, ∞), where the use of parentheses indicates that infinity is not a number that can be reached or included in the set, but rather a concept that extends beyond any finite boundary. Similarly, the notation \([a, b]\) would mean all numbers from 'a' to 'b', including 'a' and 'b'. In contrast, \((a, b]\) includes numbers from 'a' to 'b' but excludes 'a' while including 'b'. Understanding how to read and write in interval notation is essential for math students, especially when dealing with the domain and range of functions.
Real Numbers
The set of real numbers encompasses all the numbers that can be found on the number line. This includes not only the integers and fractions but also irrational numbers such as \(\sqrt{2}\) or \(\pi\). In essence, any number that can represent a quantity along an endless straight line qualifies as a real number.
Real numbers are divided into several subcategories, including rational numbers (which can be expressed as a fraction of two integers), irrational numbers (which cannot be expressed as a simple fraction), and even complex numbers when they have zero imaginary parts. In the context of the exercise, when we refer to the 'domain of a function' as 'all real numbers' we are implying that there is no restriction; the function can take any value along the infinite continuum of the number line, thus supporting the domain denoted by the interval notation (-∞, ∞).
Real numbers are divided into several subcategories, including rational numbers (which can be expressed as a fraction of two integers), irrational numbers (which cannot be expressed as a simple fraction), and even complex numbers when they have zero imaginary parts. In the context of the exercise, when we refer to the 'domain of a function' as 'all real numbers' we are implying that there is no restriction; the function can take any value along the infinite continuum of the number line, thus supporting the domain denoted by the interval notation (-∞, ∞).
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