Question

Let \( [x] \) represent the greatest integer not more than \( x \). The discontinuous points of the function \[ f(x) = \frac{5 + [x]}{\sqrt{11 + [x]} - 6x + 2 + [x]} \] lie in the interval:

• A. \( [0, \infty) \)
• B. \( [5, 8] \)
• C. \( [7, 8] \)
• D. \( [7, 10] \)

Hint:

For functions involving greatest integer functions and square roots, first solve for where the denominator equals zero and check for any undefined behavior within the range.

Answer 1

The Correct Option is C

We need to identify the discontinuities of the given function. The discontinuities typically occur when the denominator equals zero or the expression inside the square root becomes negative. Step 1: Analyze the denominator \( \sqrt{11 + [x]} - 6x + 2 + [x] \). Step 2: Find the points where the denominator becomes zero and solve for \( x \). Step 3: After calculating, the discontinuous points lie in the interval \( [7, 8] \). % Final Answer The discontinuous points of the function lie in the interval \( [7, 8] \).