91Ó°ÊÓ

Rational functions are fractions where both the numerator and the denominator are polynomials. These functions can be represented in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial functions. The key feature of rational functions is that they are continuous everywhere they are defined. This means they are smooth and have no breaks except at specific points, which occur when the denominator equals zero.
Rational functions are extremely versatile and appear in many real-world scenarios. Understanding them is crucial as they often model situations involving rates, such as speed or density. However, due to the condition that the denominator cannot be zero, special attention must be given to finding these points to fully understand their behavior.

A point of discontinuity occurs in a rational function when its denominator equals zero. At these points, the function is undefined because division by zero is mathematically impossible.

• To find points of discontinuity, set the denominator of the rational function to zero and solve for \(x\).
• After finding these values, you know where the function "breaks" or is not continuous.

In the example exercise, the denominator \(x^2 - 3x - 10\) was set to zero. Solving this equation helps to find the exact points of discontinuity, which are \(x = 5\) and \(x = -2\). These are the only points where the function is not smooth and continuous.

Quadratic equations are polynomial equations of degree two, typically written as \(ax^2 + bx + c = 0\). Solving these equations is essential in understanding rational functions, especially when determining points of discontinuity.
There are multiple methods to solve quadratic equations:

• Factoring: Express the quadratic in a product form, such as \((x - p)(x - q) = 0\), and find solutions \(x = p\) and \(x = q\).
• Quadratic Formula: Use the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) when factoring is not straightforward.
• Completing the Square: Convert the equation to a perfect square trinomial form to find the solution.

In the given exercise, the factorization method was used to solve the quadratic \(x^2 - 3x - 10 = 0\), yielding the roots \(x = 5\) and \(x = -2\). Understanding how to solve these equations helps identify where the rational function is discontinuous.