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A rational function is a type of mathematical expression that represents the division of two polynomials. It is in the form of \( \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). The key characteristic of rational functions is that they can be simplified into fractions, and just like fractions in basic arithmetic, they are undefined when their denominator is zero.

For a rational function like \( f(x)=\frac{1}{x+7}+\frac{3}{x-9} \), we're dealing with two separate fractions. Here, the denominators are \( x+7 \) and \( x-9 \) which means that the value of \( x \) cannot make either of these denominators zero to avoid undefined expressions. Thus, understanding rational functions heavily relies on figuring out where the function can and cannot have values, which leads us to the concept of a function's domain.

The concept of undefined points in functions is important when determining the domain, especially for rational functions. These are specific values of \( x \) that would cause the function to have a denominator of zero, leading to an undefined expression. Since division by zero is not allowed in mathematics, we must exclude these values from the domain.

In the exercise \( f(x)=\frac{1}{x+7}+\frac{3}{x-9} \), the undefined points are found by setting the denominators equal to zero and solving for \( x \). This gives us \( x = -7 \) and \( x = 9 \) as the undefined points for \( f(x) \). It's critical when working with functions to always check for these values to ensure the function is properly defined over its domain.

The process of solving equations is fundamental in mathematics. When determining the domain of a function, particularly a rational one, we often have to solve simple equations to find the undefined points. Solving equations typically involves isolating the variable we are interested in; in most cases, this is \( x \).

For example, when we have \( x + 7 = 0 \), we solve for \( x \) by subtracting 7 from both sides, thus \( x = -7 \). Similarly, \( x - 9 = 0 \) is solved by adding 9 to both sides, leading to \( x = 9\). These solutions are the values that cannot be included in the domain of \( f(x) \) because they would make the function undefined. Knowing how to solve such equations is essential for understanding the behavior of functions and determining where they exist.