Question

The function \[ f(x) = \frac{x + 1}{9x + x^3} \] is

• A. discontinuous at exactly two points.
• B. continuous for all real values of \( x \).
• C. discontinuous at exactly three points.
• D. discontinuous at exactly one point.

Hint:

To find points of discontinuity in rational functions, set the denominator equal to zero and solve for \( x \).

Answer 1

The Correct Option is D

Step 1: Identify the points of discontinuity.
We are given the function \( f(x) = \frac{x + 1}{9x + x^3} \). The function is undefined where the denominator is zero, i.e., when \( 9x + x^3 = 0 \). Solving this, we find \( x = 0 \) as the only point of discontinuity.

Step 2: Conclusion.
Thus, the function is discontinuous at exactly one point, corresponding to option (D).