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Rational functions are a type of function that can be expressed as the quotient of two polynomials. In simpler terms, it's like a fraction where the top part (numerator) and the bottom part (denominator) are both polynomial expressions. The general form for a rational function is \( f(x) = \frac{P(x)}{Q(x)} \). Here, \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero. If \( Q(x) \) were zero, the expression would be undefined because division by zero is not allowed in mathematics.

When solving problems involving rational functions, like finding the domain, we pay special attention to the denominator. The domain consists of all possible real values of \( x \) except those that make \( Q(x) = 0 \). These problematic values are excluded because they would make the function undefined. Therefore, the key task is to find these values and exclude them from the domain.

The difference of squares is a specific algebraic pattern that involves two squared terms with a subtraction sign between them. The formula for the difference of squares is \( a^2 - b^2 = (a-b)(a+b) \). It's a powerful tool often used to simplify expressions or solve equations.

In the exercise, the denominator of the function \( x^2 - 49 \) is a difference of squares. Here, \( a = x \) and \( b = 7 \), because \( 49 = 7^2 \). By applying the difference of squares formula, \( x^2 - 49 \) can be factored into \((x - 7)(x + 7)\). This factoring helps in determining which values will make the denominator zero. It's a commonly used technique in algebra to make solving equations more manageable.

Interval notation is a mathematical syntax used to describe subsets of real numbers, particularly useful for expressing domains. It provides a concise way of indicating which values are included and excluded from a set.

For the rational function in the exercise, the values of \( x \) that make the denominator zero, namely \( -7 \) and \( 7 \), need to be excluded from the domain. The domain in interval notation is written as:

• \((-\infty, -7) \cup (-7, 7) \cup (7, \infty)\)

Here,
- \((-\infty, -7)\) represents all numbers less than \(-7\),
- \((-7, 7)\) represents all numbers between \(-7\) and \(7\), excluding both endpoints, and
- \((7, \infty)\) represents all numbers greater than \(7\).

By using interval notation, it's easy to see that \(-7\) and \(7\) are the only values not included in the domain of the function.