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Consider the function f (x) = StartFraction x squared minus 12 x + 20 Over x minus 2 EndFraction. Which statement describes whether the function is continuous at x = 2?

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Final answer:

The function f(x) is continuous at x = 2 because upon factoring and simplifying the expression, the problematic factor cancels out, leaving a linear function which is continuous, and the limit at x = 2 matches the simplified function's value.

Step-by-step explanation:

The student's question involves determining if the function f(x) = \frac{x^2 - 12x + 20}{x - 2} is continuous at x = 2. To find out, we need to analyze the behavior of the function as it approaches the value x = 2.

First, we try to simplify the function. Factor the numerator to get (x - 10)(x - 2). Then, we notice that the factor (x - 2) in the numerator cancels out with the denominator, assuming x \\eq 2. What remains is f(x) = x - 10, which is continuous everywhere.

Since the point of contention is x = 2, we check the limit of f(x) as x approaches 2. After factoring and canceling, we are left with the limit of x - 10, which as x approaches 2, approaches -8. Therefore, we can say that the function is continuous at x = 2, because the limit exists and the value of the simplified function at x = 2 is -8, matching the limit.

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