91Ó°ÊÓ

Rational functions are expressions that can be written as the ratio of two polynomials. They have the general form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). The function \( y = \frac{x+2}{\cos x} \) is a rational function, because it is structured as a ratio, albeit with a trigonometric denominator.
These types of functions have unique characteristics:

• The domain includes all real numbers except where the denominator equals zero.
• The graph can have vertical asymptotes, which occur where the denominator is zero.
• Rational functions can also have horizontal asymptotes, determined by the degrees of the numerator and denominator polynomials.

Understanding these characteristics helps identify the points of discontinuity in the function, particularly by analyzing where the denominator is zero.

Trigonometric functions, such as sine, cosine, and tangent, are essential in describing cyclical patterns. In the given function \( y = \frac{x+2}{\cos x} \), the cosine function \( \cos x \) appears in the denominator, playing a crucial role in determining where the function is undefined.
Key characteristics of the cosine function are:

• Cosine values oscillate between \(-1\) and \(1\).
• It reaches zero at \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
• Because \( \cos x \) can be zero, it influences where the function \( y \) is discontinuous.

By understanding these aspects, we can identify that the continuities and discontinuities of the original function are based around these trigonometric properties.

An expression becomes undefined when it involves a division by zero. This occurs in rational functions where the denominator equals zero. For the function \( y = \frac{x+2}{\cos x} \), the same rule applies. To identify the undefined points of this function, it's crucial to determine when \( \cos x = 0 \).
Cosine becomes zero at odd multiples of \( \frac{\pi}{2} \). Therefore, \( x = \frac{\pi}{2} + n\pi \) represents values where the function becomes undefined. This condition shows that the set of all integers \( n \) affects the points of discontinuity.
Identifying undefined expressions helps us understand why rational functions are not valid at certain points and allows us to pinpoint where discontinuities occur due to division by zero.