Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Find the domain of the expression. \(\sqrt{81-4 x^{2}}\)

Step-by-Step Solution

Answer

The domain of the expression \(\sqrt{81-4 x^{2}}\) is \([-4.5, 4.5]\).

Step 1: Set the Condition for the Square Root Function

In the real number system, a square root function is only defined when the expression under the square root, or radicand, is greater than or equal to zero. So, set the radicand greater than or equal to zero. \[81-4 x^{2} \geq 0\]

Step 2: Rearrange the Equation

Rearrange the equation to isolate the x term. Divide both sides of the inequality by -4, remember to flip the inequality because you are dividing by a negative number, yielding: \[-x^{2} \geq -20.25\]

Step 3: Solve for x

Take the square root of both sides (remembering to consider both the positive and negative roots) to yield \[x \leq 4.5\] or \[x \geq -4.5\]

Step 4: State the Domain

The numbers that satisfy the inequality form the domain of the function. The answer is expressed in interval notation as:\[-4.5 \leq x \leq 4.5\] or in interval notation \([-4.5, 4.5]\)

Key Concepts

Square Root Function

To understand the domain of an expression that includes a square root, it's crucial to first grasp what a square root function entails. A square root function is written as \( f(x) = \sqrt{x} \) and involves taking the square root of a variable, say \( x \). The crucial thing to remember is that the square root is only real and defined for non-negative numbers, meaning the domain of \( f(x) = \sqrt{x} \) includes all \( x \geq 0 \).

In the context of our problem, the expression under the square root is \( 81 - 4x^2 \), referred to as the radicand. To find values of \( x \) that keep the radicand non-negative, we set up the inequality \( 81 - 4x^2 \geq 0 \). Solving this inequality will give us the range of \( x \) for which the square root function is defined, thus determining the domain.

Inequalities in Algebra

Inequalities are a staple in algebra and are used to describe the range of values that satisfy a particular condition. They come in various forms, such as \( <, \leq, >, \geq \). When solving inequalities, especially those involving a square root function, we seek to determine the set of all possible values that make the inequality true.

In the given exercise, we encounter the inequality \( 81 - 4x^2 \geq 0 \). The process requires manipulating the inequality so that we can isolate \( x \). Caution must be taken when dividing by negative numbers as it requires reversing the inequality symbol. In our case, dividing by -4 flips \( \geq \) to \( \leq \), leading to \( -x^2 \geq -20.25 \) and subsequently finding that \( x \leq 4.5 \) and \( x \geq -4.5 \). This represents the solutions for \( x \) which keep the radicand non-negative, vital for determining the expression's domain.

Interval Notation

Interval notation is a neat and compact way of expressing a range of values satisfying an inequality or condition. It is particularly useful in defining domains and ranges of functions in mathematics. In interval notation, square brackets, \( [ \) and \( ] \), indicate that an endpoint is included, also known as a closed interval, whereas parentheses, \( ( \) and \( ) \), indicate that an endpoint is not included, known as an open interval.

For the solution of our inequality, the domain of the square root function is from -4.5 to 4.5, inclusive of both endpoints, because we deal with a \( \leq \) inequality. Therefore, in interval notation, we write this as \( [-4.5, 4.5] \). This notation succinctly conveys that every number between -4.5 and 4.5, including the endpoints, is a part of the domain for the given square root expression.