Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-32: Find the domain of \(f\). $$ f(x)=\frac{1}{(x-3) \sqrt{x+3}} $$

Step-by-Step Solution

Answer

The domain of \(f(x)\) is \((-3, 3) \cup (3, \infty)\).

Step 1: Identify Restrictions on Division

To find the domain of a function, we must first ensure the denominator is not equal to zero because division by zero is undefined. For \( f(x) = \frac{1}{(x-3) \sqrt{x+3}} \), both the term \( x-3 \) and \( \sqrt{x+3} \) affect the denominator.

Step 2: Set Denominator Conditions

We need \((x-3)\sqrt{x+3} eq 0\). This means neither \(x-3 = 0\) nor \(\sqrt{x+3} = 0\). \(x-3 = 0\) gives \(x = 3\) as a restriction. \(\sqrt{x+3} = 0\) means \(x+3 \geq 0\), hence \(x \geq -3\) must hold, eliminating negative values and ensuring the square root is defined.

Step 3: Combine Restrictions

From Steps 1 and 2, we combine these insights. \(x\) must be greater than \(-3\) and cannot be equal to \(3\). This analysis gives the domain restriction as \(x > -3 \text{ and } x eq 3\).

Step 4: Write the Domain

In interval notation, the domain of \(f(x)\) is \((-3, 3) \cup (3, \infty)\), representing all real numbers greater than \(-3\) except 3, where the function is undefined.

Key Concepts

Function Restrictions

Understanding function restrictions is crucial when determining the domain of functions, especially those involving fractions or square roots. In simple terms, function restrictions are conditions that limit the values a variable can take in a function. These restrictions are essential to prevent cases where the function becomes undefined, such as division by zero or taking the square root of a negative number. In our example, the function is given by \[ f(x) = \frac{1}{(x-3) \sqrt{x+3}} \].This expression means that we must find values of \(x\) that do not cause the denominator to become zero or the square root to be undefined. By identifying these restrictions, we set the groundwork for finding the domain—the set of all possible \(x\) values that make the function valid.

Interval Notation

Interval notation is a concise way of describing sets of numbers, typically representing domains of functions. It's an efficient tool to express which numbers are included in a function's domain and which are not. In interval notation:

• Round brackets \(( \text{or } )\) indicate that a boundary is not included, known as open intervals.
• Square brackets \([ \text{or } ]\) indicate that a boundary is included, known as closed intervals.

For the function \(f(x) = \frac{1}{(x-3) \sqrt{x+3}}\), the domain is given as \((-3, 3) \cup (3, \infty)\).This notation shows that all real numbers greater than \(-3\) but not equal to 3 are included in the domain, joining two segments where the function remains valid.

Division by Zero

Division by zero is one of the fundamental restrictions when dealing with fractions in functions. It occurs when the denominator of a fraction becomes zero, leading to an undefined expression. To avoid this, we must ensure that the denominator never equals zero for any value of the variable.In the function\[ f(x) = \frac{1}{(x-3) \sqrt{x+3}} \],the denominator is \((x-3) \sqrt{x+3}\).Here, the term \(x-3\) must not equal zero, which implies that \(x eq 3\). Hence, \(x = 3\) is excluded from the domain to prevent division by zero. Understanding this concept is essential to correctly specify the domain of any function with fractional components.

Square Root Inequality

The square root inequality is another vital consideration when determining the domain of a function. It ensures the expression under a square root is non-negative because the square root of a negative number is not defined within the set of real numbers.For our function \( f(x) = \frac{1}{(x-3) \sqrt{x+3}} \), we consider \(\sqrt{x+3}\).The expression inside the square root, \(x+3\), must be greater than or equal to zero, requiring \(x \geq -3\). This step assures that the square root operation is defined and valid. Combining this with the restriction from division by zero, we refine our domain to values of \(x\) greater than \(-3\) but not equal to 3, ensuring the function can operate correctly without encountering undefined values.