Question

The number of point / points where the function \(f(x) = \frac{1}{x^2-5|x|+6}\) is discontinuous is......

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(4\)

Hint:

The function is discontinuous where the denominator is zero.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A rational function is discontinuous at points where the denominator is zero. Here the denominator is \(x^2 - 5|x| + 6\).

Step 2: Key Formula or Approach:
Since \(x^2 = |x|^2\), put \(t = |x|\): \(t^2 - 5t + 6 = (t - 2)(t - 3)\).

Step 3: Detailed Explanation:
Zero when \(|x| = 2\) or \(|x| = 3\).
That gives \(x = \pm 2\) and \(x = \pm 3\), which are \(4\) different points.
No common factor cancels with the numerator, which is \(1\), so each of these four points is a real discontinuity.
The function is continuous everywhere else.

Final Answer:
There are \(4\) points of discontinuity, option (D). \[ \boxed{4} \]