Question

The function \( f(x) = \frac{8x}{x^2 + 9} \) is continuous everywhere except at

• A. \( x = 0 \)
• B. \( x = \pm 9 \)
• C. \( x = \pm 9i \)
• D. \( x = \pm 3i \)

Hint:

For rational functions, check where the denominator is zero to identify points of discontinuity. In this case, the denominator is always positive for real values of \( x \).

Answer 1

The Correct Option is D

Step 1: Understanding the function.
The function given is: \[ f(x) = \frac{8x}{x^2 + 9} \] This is a rational function, and it will be continuous wherever the denominator is not equal to zero.
Step 2: Checking for discontinuities.
The denominator \( x^2 + 9 \) is never zero for real values of \( x \). The quadratic expression \( x^2 + 9 \) has no real roots, so the function is continuous for all real values of \( x \).
Step 3: Conclusion.
The only point where the function could potentially be undefined is for complex values of \( x \), but the function is continuous everywhere except at \( x = 0 \) for real values. Hence, the correct answer is \( x = 0 \).