Problem 64 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=\frac{x}{\sqrt[4]{9-x^{2}}} $$

Step-by-Step Solution

Answer

The domain of the function is \((-3, 3)\).

Step 1: Understand the Problem

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For this function, we need to ensure both the numerator and denominator are defined and that the denominator is not zero.

Step 2: Analyze the Denominator

The denominator is \( \sqrt[4]{9-x^2} \). This expression must be greater than zero because a root cannot be zero in the denominator of a fraction. Thus, we need:\[9 - x^2 > 0\]

Step 3: Solve the Inequality

Solve the inequality \( 9 - x^2 > 0 \). First rewrite it as:\[9 > x^2\]This can be expressed as:\(-3 < x < 3\).

Step 4: Write the Domain

The domain of the function is all x-values for which \(-3 < x < 3\). In interval notation, this is written as \((-3, 3)\).

Key Concepts

Function Analysis

Understanding the domain of a function is a crucial part of function analysis. The domain represents all the possible input values, represented as "x-values," for which a given function is mathematically valid and defined. When analyzing functions such as \( f(x) = \frac{x}{\sqrt[4]{9-x^2}} \), it is essential to ensure that each part of the function can operate without error.
For any rational function, where a ratio of expressions is present, the denominator cannot be zero because division by zero is undefined. Moreover, for any radical expressions, such as square roots or fourth roots, the expression inside must be non-negative to ensure it results in a real number, as complex numbers are often beyond the scope of such analyses. By focusing on these aspects, you can determine valid x-values that define the domain.
Start by checking conditions that arise from the denominator and radical expressions, as these often exclude certain x-values, shaping the domain of the function.

Inequality Solving

To find where a function is defined, you might need to delve into inequality solving. This ensures certain expressions, especially those restricting certain values, remain within suitable ranges to keep the function valid. Consider the function \( f(x) = \frac{x}{\sqrt[4]{9-x^2}} \). The presence of a root in the denominator means the expression inside must remain positive, so \(9 - x^2 > 0\).
Solving such inequalities involves finding all permissible x-values.

• Rearrange the inequality: For \(9 - x^2 > 0\), rewrite it as \( x^2 < 9 \).
• Solve the inequality: This tells us \(-3 < x < 3\), meaning x must be less than 3 and more than -3.

By accurately solving these inequalities, you determine the full set of x-values where the function stands defined, playing a vital role in identifying the domain.

Interval Notation

Once the valid range of x-values is identified using inequalities, interval notation succinctly expresses the domain. This mathematical shorthand helps easily communicate the set of numbers satisfying the inequalities and, in this context, the domain of the analyzed function.
For \( f(x) = \frac{x}{\sqrt[4]{9-x^2}} \), after solving and understanding that \(-3 < x < 3\), we translate this into interval notation: \((-3, 3)\).
The rounded brackets \(( )\) suggest that \(-3\) and \(3\) are not included in the domain, while square brackets \([ ]\) would indicate inclusion. Using such notation is a clear and concise way to represent the domain, allowing other mathematicians or students to understand the valid inputs quickly.
Mastering interval notation not only aids with domains but also plays a role in various areas of mathematics involving ranges and solutions of inequalities.