Problem 21 - Thomas Calculus

Question

Find the domain of $$y=\frac{x+3}{4-\sqrt{x^{2}-9}}$$

Step-by-Step Solution

Answer

The domain is \( (-\infty, -5) \cup (-5, -3] \cup [3, 5) \cup (5, \infty) \).

Step 1: Identify the Denominator

The function given is \( y = \frac{x+3}{4-\sqrt{x^2-9}} \). The denominator is \( 4 - \sqrt{x^2-9} \). To determine the domain, we need to ensure that the denominator is not equal to zero.

Step 2: Set the Denominator Not Equal to Zero

Set \( 4 - \sqrt{x^2-9} eq 0 \). This simplifies to \( \sqrt{x^2-9} eq 4 \). By squaring both sides, we get \( x^2 - 9 eq 16 \). Thus, \( x^2 eq 25 \), giving us \( x eq 5 \) and \( x eq -5 \).

Step 3: Consider Condition for Square Root

For \( \sqrt{x^2-9} \) to be defined, the expression inside the square root must be non-negative. Thus, \( x^2 - 9 \geq 0 \). Solving this, we get two intervals: \( x \geq 3 \) or \( x \leq -3 \).

Step 4: Combine Conditions

Combine the conditions from the previous steps. The square root condition gives us the intervals \( x \geq 3 \) or \( x \leq -3 \). Additionally, \( x eq 5 \) and \( x eq -5 \). Thus, the domain is \( x \leq -3 \) excluding \( x = -5 \), and \( x \geq 3 \) excluding \( x = 5 \).

Step 5: Express the Domain in Interval Notation

The domain in interval notation is \( (-\infty, -5) \cup (-5, -3] \cup [3, 5) \cup (5, \infty) \). This includes all values where the function is defined, considering both the denominator non-zero and the valid square root condition.

Key Concepts

Domain of a Function

A domain in mathematics specifies all the possible input values (usually represented by "x") for a function. These are the values that make the function work without any math errors, like division by zero or the square root of a negative number. For our function, \( y = \frac{x+3}{4-\sqrt{x^2-9}} \), we need to find all possible \( x \) values where the function is defined.

To achieve this, it is important to ensure:

• The denominator of the function does not equal zero, since division by zero is undefined.
• Sensible calculations exist for other operations involved, such as taking square roots.

By combining these considerations, we will define the domain of the function accurately.

Square Root Properties

The square root operation, denoted by \( \sqrt{\cdot} \), must only operate on non-negative numbers in real number calculations. This means inside of a square root, any expression must be zero or positive.

For example, for \( \sqrt{x^2-9} \) in our function, the term \( x^2-9 \) must satisfy \( x^2-9 \geq 0 \) to produce real number results. Solving \( x^2-9 \geq 0 \) gives us conditions for \( x \), telling us that \( x \geq 3 \) or \( x \leq -3 \).

Therefore, when working with square roots:

• Ensure the value under the root is zero or positive.
• Identify intervals where the square root is defined based on conditions satisfied by the expression inside the root.

Interval Notation

Interval notation offers a compact way to showcase the set of numbers forming the domain, helping describe continuous parts of the domain efficiently.

It uses brackets and parenthesis:

• "[ ]" indicates that the endpoints are included in the set (closed interval).
• "( )" indicates that endpoints are not included in the set (open interval).

In our function's domain:

- The intervals \( (-\infty, -5) \cup (-5, -3] \cup [3, 5) \cup (5, \infty) \) are established, where "\(-\infty, -5\)" and "\(5, \infty\)" show certain x-values are not included, adhering to square root conditions or non-zero denominators. This notation efficiently communicates the domain, allowing immediate understanding of which x-values can be used.

Exclusion of Values

When determining the domain, sometimes specific values of \( x \) must be excluded to avoid issues such as undefined operations. For our function, \( y = \frac{x+3}{4-\sqrt{x^2-9}} \), we identified points \( x = 5 \) and \( x = -5 \) that cause the denominator to become zero, so they must be excluded from the domain.

This exclusion is crucial for ensuring correct mathematical operations:

• Identify points at which an operation, like division, leads to undefined results.
• Exclude these values from the domain to maintain proper function definition and calculation integrity.

By excluding \( x=5 \) and \( x=-5 \), we ensure that unwanted discontinuities do not occur, preserving the accurate representation of the function's behavior.