Questions: Determine analytically the domain of the function below and write your answer in interval notation. h(x) = sqrt((x+4)/(x-7))

Solution Steps

To determine the domain of the function \( h(x) = \sqrt{\frac{x+4}{x-7}} \), we need to ensure that the expression inside the square root is non-negative and the denominator is not zero. This means solving the inequality \(\frac{x+4}{x-7} \geq 0\) and ensuring \(x \neq 7\).

The function given is

\[ h(x) = \sqrt{\frac{x+4}{x-7}} \]

To determine the domain of this function, we need to consider the restrictions imposed by the square root and the fraction.

1. Square Root Restriction: The expression inside the square root must be non-negative:

   \[ \frac{x+4}{x-7} \geq 0 \]

2. Denominator Restriction: The denominator cannot be zero:

   \[ x - 7 \neq 0 \quad \Rightarrow \quad x \neq 7 \]

To solve the inequality \(\frac{x+4}{x-7} \geq 0\), we need to find the critical points where the expression is zero or undefined:

• Zero of the Numerator: \(x + 4 = 0 \Rightarrow x = -4\)
• Zero of the Denominator: \(x - 7 = 0 \Rightarrow x = 7\)

These critical points divide the number line into intervals. We will test each interval to determine where the inequality holds.

The critical points divide the number line into the following intervals: \((- \infty, -4)\), \((-4, 7)\), and \((7, \infty)\).

1. Interval \((- \infty, -4)\):

   • Choose \(x = -5\): \[ \frac{-5+4}{-5-7} = \frac{-1}{-12} = \frac{1}{12} > 0 \] The inequality holds.

2. Interval \((-4, 7)\):

   • Choose \(x = 0\): \[ \frac{0+4}{0-7} = \frac{4}{-7} < 0 \] The inequality does not hold.

3. Interval \((7, \infty)\):

   • Choose \(x = 8\): \[ \frac{8+4}{8-7} = \frac{12}{1} = 12 > 0 \] The inequality holds.

• At \(x = -4\), \(\frac{x+4}{x-7} = 0\), which satisfies the inequality.
• At \(x = 7\), the expression is undefined, so \(x = 7\) is not included in the domain.

Final Answer