Question

Determine the intervals on which the following function is continuous f(x)=(x-10)/(x²-100)

A. (-[infinity], -10) ∪ (10, [infinity])
B. (-10, -√100) ∪ (-√100, 10) ∪ (10, √100) ∪ (√100, [infinity])
C. (-√100, √100)
D. (-[infinity], 10) ∪ (-10, [infinity])
E. (-10, √100) ∪ (√100, 10)

Answer 1

The function f(x) = (x - 10) / (x² - 100) is continuous on the intervals (-∞, -10) and (10, ∞), with discontinuities at x = -10 and x = 10, matching option A.

Step-by-step explanation:

To determine the intervals on which the function f(x) = (x - 10) / (x² - 100) is continuous, we need to consider the points where the function could be undefined. The denominator x² - 100 can be factored into (x + 10)(x - 10). Thus, the function is undefined at x = -10 and x = 10, because division by zero is not allowed in mathematics.

Aside from these points, the function is defined for all other real numbers. Therefore, the function is continuous on the intervals (-∞, -10) and (10, ∞), which can be expressed as the union of these two intervals. This corresponds to option A:

(-∞, -10) ∪ (10, ∞)

In summary, the continuous intervals for the function f(x) = (x - 10) / (x² - 100) do not include the points where the function is undefined due to division by zero, resulting in two separate intervals on the real number line.