91Ó°ÊÓ

Rational functions are a type of algebraic function, defined as the quotient of two polynomials. In simpler terms, they look like fractions where both the numerator and the denominator are polynomials. The function given in the problem, \[g(x) = x + \frac{3}{5-x}\]is a rational function because of the fraction \(\frac{3}{5-x}\). Understanding the behavior of rational functions relies significantly on examining the denominator. This is important because the values of the variable, \(x\), that make the denominator zero, are problematic as division by zero is undefined in mathematics. Such values are not included in the function's domain.
Identifying these values is the first step to solving problems associated with rational functions. Thus, a key aspect of analyzing rational functions involves determining where the function is not defined and excluding those points from the domain.
Since the domain greatly affects the real-world applicability of functions, in many mathematical applications, recognizing restrictions in rational functions is essential.

Inequalities allow us to determine the range or domain of a function by setting limits on the values that variables can take. In the context of rational functions like \(g(x) = x + \frac{3}{5-x}\), identifying inequalities helps to figure out the set of values excluded from the domain.
To find which values of \(x\) make the denominator zero, we set an inequality ensuring that it does not equal zero. As worked out in the solution: \[5 - x eq 0\]Solving this inequality:

• Subtract \(5\) from both sides: \[-x eq -5\]
• Multiply both sides by \(-1\) to isolate \(x\):\[x eq 5\]

This systematic process ensures that \(x\) cannot be 5, and thereby keeps the function defined.
Working through inequalities in rational functions is necessary to properly describe intervals where the function holds true and where it doesn't, rendering such practice invaluable.

Interval notation is a method used to represent the set of all values that form a domain or solution to inequalities. It helps specify precisely where different criteria such as restrictions and inequalities apply. In the original exercise, we found that the domain of the function excluded the value \(x = 5\).
Using interval notation, this domain is expressed as:\[(-\infty, 5) \cup (5, \infty)\]
This notation tells us:

• \((-\infty, 5)\): includes all real numbers less than 5.
• \((5, \infty)\): includes all real numbers greater than 5.
• The "\(\cup\)" symbol signifies a union, which combines the two intervals.

Interval notation is both a precise and concise method of representing domains, especially important in higher mathematics or when conveying the solution to functions like the one in the exercise. It helps mathematicians communicate complex ideas clearly, reducing ambiguity about which values are part of the solution set.