Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Find the domain of the expression. \(\sqrt[4]{6+x^{2}}\)

Step-by-Step Solution

Answer

The domain of the expression \(\sqrt[4]{6+x^{2}}\) is all real numbers, or in interval notation: \(-\infty , +\infty\)

Step 1: Set up the inequality

First, set the condition of what's inside the root greater than or equal to 0. This leads to: \(6 + x^2 \geq 0\).

Step 2: Analyze the inequality

Note that this inequality will hold true for \(-\sqrt{6} \leq x \leq \sqrt{6}\), where \(\sqrt{6}\) is a positive root of the quadratic equation \(x^2 = -6\). However, the quadratic expression \(6 + x^2\) is always non-negative because \(x^2\) is non-negative for all real values of x. The term '6' just shifts it 6 units upward. This implies that the domain of the expression \(\sqrt[4]{6 + x^2}\) exists for all real numbers.

Step 3: Formulate the domain

After analysing, write the domain as: All real numbers, which can also be written in interval notation as: \(-\infty , +\infty\)

Key Concepts

Inequalities

Understanding inequalities is a fundamental concept in mathematics, especially when determining the domain of an expression. An inequality indicates a relationship in which one value is not equal to another; it can be less than (<), greater than (>), less than or equal to () or greater than or equal to (). For the given exercise, the domain of the radical expression is found by setting up an inequality: ). For the given exercise, the domain of the radical expression is found by setting up an inequality: ). For the given exercise, the expression determines the domain of the radical expression. The domain of the radical expression is found by setting up an inequality: ). For the given exercise, the domain of the radical expression is found by setting up an inequality: ). For the given exercise, the domain of the radical expression is found by setting up an inequality: ). For the given exercise, the domain of the radical expression is found by setting up an inequality: ). For the given exercise, the domain of the radical expression is found by setting up an inequality: ). For the given exercise, the domain of the radical expression is defined by the setting up an inequality: ). For the given exercise, the domain of the radical expression is defined by the setting up an inequality: ). For the given exercise, the domain of the radical expression is defined by the setting up an inequality: ). For the given exercise, the domain of the radical expression is defined by the setting up an inequality: ). For the given exercise, the domain of the radical expression is defined by the setting up an inequality: ). For the given exercise, the domain of the radical expression is defined by the setting up an inequality: ). For the given exercise, the domain of the radical expression is determined by setting up an inequality: ). For the given exercise, the domain of the radical expression is defined by the setting up an inequality: ). For the given exercise, the domain of the radical expression is defined by the setting up an inequality: ). To ascertain the domain, one must solve the inequality and ensure it corresponds to the conditions of the given mathematical operations – in this case, finding a real number range where the fourth root is defined.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. When dealing with roots, especially even-indexed roots like square roots or fourth roots, it is critical to ensure the expression under the root is non-negative, as even roots of negative numbers are not real. This principle applies directly to our problem where the expression is ). The expression's domain is affected as negative inputs for even-indexed radical expressions could lead to non-real numbers. It is not the case with odd-indexed radicals since they allow for negative inputs. Thus, for the given ), a real result for all real values of x because squaring any real number always results in a non-negative value. This concept is vital to understanding the domain of functions involving radicals.

Quadratic Equations

Quadratic equations take the form ), where ), ), and ) are constants, with ). These equations play a role in solving inequalities that involve a square root or a radical with an even index, as they can determine the values for which the radicand (the expression under the root) is non-negative. This turns out to be a pivotal step in determining the domain of a radical expression. In our exercise, the presence of the term ) is a clear indication of a quadratic expression. Although not explicitly solved as a quadratic equation, the inequality ) reflects the principle that a parabola defined by ) where ) is positive, never dips below the x-axis, ensuring that all real numbers are within the domain.

Interval Notation

Interval notation is a standardized way to represent subsets of the real number line. It employs brackets and parentheses to indicate closed and open intervals, respectively, with closed intervals [], ] including the endpoints and open intervals (, ) not including the endpoints. Mathematicians favor this notation for its efficiency in conveying a range of solutions, such as the domain of a function. In the solution provided, the domain for the expression ) is all real numbers, indicated by the interval ). It succinctly communicates that there is no real number which can’t be plugged into the given radical expression, thus making every real number a part of the function's domain. Understanding interval notation is essential for clearly presenting the results found through solving inequalities, radical expressions, and other mathematical problems.