Determine the domain $$ f(x) = \frac{x + 2}{\sqrt{x^2 - 9}} $$

Domain of a Rational Function with a Square Root

In this lesson, we will learn how to determine the domain of a rational function that contains a square root in the denominator. Finding the domain is an important step because it tells us which values of x make the function valid.

1. Theory of Domain

The domain of a function is the set of all input values for which the function is defined. For rational functions with square roots, special rules apply. We must make sure that the denominator is not zero and that any expression inside a square root is positive.

2. Concept Explanation

When a square root appears in the denominator, the expression inside the square root must be greater than zero. This is more restrictive than simply avoiding zero in the denominator. By solving the resulting inequality, we can determine which values of x are allowed.

3. Steps or Formula

1. Write down the given function.
2. Ensure the denominator is not equal to zero.
3. Require the expression inside the square root to be greater than zero.
4. Solve the resulting inequality.
5. Express the domain using set notation and interval notation.

4. Example Problem

Given function:

$$ f(x) = \frac{x + 2}{\sqrt{x^2 - 9}} $$

Step 1: Apply the domain conditions.

The denominator cannot be zero, and the expression inside the square root must be greater than zero:

$$ x^2 - 9 > 0 $$

Step 2: Solve the inequality.

$$ x^2 - 9 = (x - 3)(x + 3) $$ $$ (x - 3)(x + 3) > 0 $$

The critical points are:

$$ x = -3 \quad \text{and} \quad x = 3 $$

The inequality is satisfied when:

$$ x < -3 \quad \text{or} \quad x > 3 $$

5. Domain Representation

On a number line, the valid values are all numbers to the left of -3 and to the right of 3. These regions represent the domain of the function.

In set notation, the domain is:

$$ D_f = \{x \in \mathbb{R} \mid x < -3 \text{ or } x > 3\} $$

In interval notation, the domain is:

$$ (-\infty, -3) \cup (3, \infty) $$

Final Answer:

The domain of the function consists of all real numbers less than -3 or greater than 3.