Question

What is the domain of the function \(f(x)=\sqrt{\frac{x-2}{x+4}} ?\)

Short answer

The domain is \( (-\infty, -4) \cup (-4, 2] \).

Step-by-step solution

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.

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 \( \frac{x-2}{x+4} \geq 0 \).

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.

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.

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.

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 \( (-\infty, -4) \cup (-4, 2] \).

Key concepts

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).

This ensures you have identified the correct domain for the square root function.

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’s 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.

This approach ensures you understand where the inequality is satisfied and thus where the square root function is defined.

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.

Understanding critical points helps you properly identify the domain of the function and ensure the square root function remains valid and defined.