Final answer:
The function Y = 2x/(x + 1) has a discontinuity at x = -1, which is an infinite discontinuity (vertical asymptote). The function Y = (x + 3)/(x² - 9) has a removable discontinuity at x = -3 and an infinite discontinuity at x = 3.
Step-by-step explanation:
The subject of the question relates to identifying points of discontinuity in given mathematical functions and classifying each type of discontinuity. Let's analyze the given functions one at a time:
For the function Y = 2x/(x + 1), there is a potential point of discontinuity where the denominator equals zero. To find it, set x + 1 = 0, which yields x = -1. Therefore, the function has a point of discontinuity at x = -1. This is a rational function, and the discontinuity is likely a vertical asymptote, which is a type of infinite discontinuity.
For the function Y = (x + 3)/(x² - 9), the points of discontinuity are again where the denominator equals zero. Factoring the denominator, we get (x + 3)(x - 3). Setting each factor equal to zero gives us x = -3 and x = 3. These values are points of discontinuity. Since x = -3 is a factor in both the numerator and denominator, it will simplify, leading to a removable discontinuity (also known as a hole), while x = 3 will lead to a vertical asymptote, which is an infinite discontinuity.
Note that these conclusions are based on algebraic analysis, and to fully classify the types of discontinuity, one should also look at the limits and behavior of the function around these points.