Final answer:
The function f(x) = (x² + 2x + 3)/(x² - x - 2) is continuous on the intervals (-∞, -1), (-1, 2), and (2, ∞), as the function is undefined at x = 2 and x = -1 where the denominator equals zero.
Step-by-step explanation:
To determine along which interval the function f(x) = (x² + 2x + 3)/(x² - x - 2) is continuous, we first need to find the values of x for which the function is undefined. This occurs when the denominator is zero, since division by zero is undefined in mathematics. Factor the denominator to find the values that make it zero:
x² - x - 2 = (x-2)(x+1)
Setting each factor equal to zero gives us the x-values where the function has discontinuities:
- x - 2 = 0 → x = 2
- x + 1 = 0 → x = -1
Therefore, the function f(x) is continuous everywhere except at x = 2 and x = -1. So, the function is continuous on the intervals (-∞, -1), (-1, 2), and (2, ∞).