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Magazine Chapter 1: Problem 21
Find the domain of $$y=\frac{x+3}{4-\sqrt{x^{2}-9}}$$
Short Answer
Expert verified
The domain is \( (-\infty, -5) \cup (-5, -3] \cup [3, 5) \cup (5, \infty) \).
Step by step solution
01
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.
02
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 \( x eq -5 \).
03
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 \( x \leq -3 \).
04
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 \( x = 5 \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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:
- 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.
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).
- 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:
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.
