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Magazine Chapter 5: Problem 63
What is the domain of the function \(f(x)=\sqrt{\frac{x-2}{x+4}} ?\)
Short Answer
Expert verified
The domain is \( (-\infty, -4) \cup (-4, 2] \).
Step by step solution
01
Understanding the function
The function given is a square root function that has a fraction inside it: \( f(x) = \sqrt{\frac{x-2}{x+4}} \). Determine the conditions where the expression inside the square root is non-negative and defined.
02
Setting the fraction non-negative
The expression inside the square root, \( \frac{x-2}{x+4} \), must be greater than or equal to zero. Therefore, solve \( \frac{x-2}{x+4} \geq 0 \).
03
Finding critical points
Determine the values that make the numerator zero (\(x-2=0\)) and the denominator zero (\(x+4=0\)). These values are \(x=2\) and \(x=-4\), respectively.
04
Creating intervals and testing
These critical points divide the number line into three intervals: \( (-\infty, -4) \), \( (-4, 2) \), and \( (2, \infty) \). Test points from each interval to check where the inequality \( \frac{x-2}{x+4} \geq 0 \) holds.
05
Considering the endpoints
Since the denominator cannot be zero, \( x = -4 \) is not included in the domain, but \( x = 2 \) can be included if it makes the expression inside the square root non-negative.
06
Conclusion
The function \( f(x) = \sqrt{\frac{x-2}{x+4}} \) is defined for \( x \) in the intervals where the expression is non-negative and the denominator is not zero. Thus, the domain is \( (-\infty, -4) \cup (-4, 2] \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Root Function
A square root function involves an expression inside a square root sign. When dealing with square root functions, it's important to ensure that the expression within the square root is non-negative. This is because the square root of a negative number is not defined within the set of real numbers. For example, in the function \(f(x)=\sqrt{\frac{x-2}{x+4}}\), the expression \(\frac{x-2}{x+4}\) must be greater than or equal to zero to ensure that the function is defined and produces real values. To determine where a square root function is defined, follow these simple steps:
- Write the inequality where the expression inside the square root is greater than or equal to zero.
- Solve the inequality to find the values of \(x\).
- Check that these values keep the denominator from being zero (if there's a fraction involved).
Inequality Solving
Solving inequalities involves determining the values of a variable that make an inequality true. In this context, the inequality stemming from the function \(f(x)=\sqrt{\frac{x-2}{x+4}}\) is \(\frac{x-2}{x+4} \geq 0\). Here鈥檚 how to solve this inequality:
- Identify the critical points by finding where the numerator and the denominator are zero. For example, \(x-2=0\) gives \(x=2\), and \(x+4=0\) gives \(x=-4\).
- Divide the number line into intervals using these critical points. In this case, the intervals are \((-\infty, -4)\), \((-4, 2)\), and \((2, \infty)\).
- Test a point from each interval in the inequality to determine where it holds true.
Critical Points
Critical points are values of \(x\) that make the numerator or the denominator of a fraction zero. They are essential for understanding the behavior of inequalities involving fractions. For the function \(f(x)=\sqrt{\frac{x-2}{x+4}}\), the critical points are \(x=2\) and \(x=-4\). These critical points divide the number line into intervals, which helps in testing where the inequality is satisfied:
- Test points from each interval: For example, test \(x=-5\) from \((-\infty, -4)\), \(x=0\) from \((-4, 2)\), and \(x=3\) from \((2, \infty)\).
- Determine if the test points satisfy the inequality \(\frac{x-2}{x+4} \geq 0\).
- Determine if the critical points themselves satisfy the inequality. For instance, \(x=2\) satisfies it because \(\frac{2-2}{2+4} = 0\), but \(x=-4\) makes the denominator zero, making the expression undefined.
