Question

If $$ f(x) = \log\left(\left(\frac{2x^2 - 3}{x}\right) + \sqrt{\frac{4x^4 - 11x^2 + 9}{|x|}}\right) $$ then $ f(x) $ is:

• A. an odd function
• B. an even function
• C. a polynomial function
• D. not a function

Hint:

To test if a function is odd, check whether \( f(-x) = -f(x) \). Odd functions are symmetric about the origin.

Answer 1

The Correct Option is A

Let us simplify the function and test its parity: Given: \[ f(x) = \log\left(\left(\frac{2x^2 - 3}{x}\right) + \sqrt{\frac{4x^4 - 11x^2 + 9}{|x|}}\right) \] Let us analyze each component: 1. The first part: \[ \frac{2x^2 - 3}{x} = \frac{2x^2}{x} - \frac{3}{x} = 2x - \frac{3}{x} \] 2. The second part: \[ \sqrt{\frac{4x^4 - 11x^2 + 9}{|x|}} \quad \text{is defined for } x \ne 0 \text{ and is even in behavior, since it involves } x^2, x^4, \text{ and } |x|. \] Now consider: \[ f(-x) = \log\left(\left(\frac{2(-x)^2 - 3}{-x}\right) + \sqrt{\frac{4(-x)^4 - 11(-x)^2 + 9}{|-x|}}\right) \] \[ = \log\left(\left(\frac{2x^2 - 3}{-x}\right) + \sqrt{\frac{4x^4 - 11x^2 + 9}{|x|}}\right) \] \[ = \log\left(-\left(\frac{2x^2 - 3}{x}\right) + \sqrt{\frac{4x^4 - 11x^2 + 9}{|x|}}\right) \] Now compare with original: \[ f(-x) = \log\left(-\left(\frac{2x^2 - 3}{x}\right) + \sqrt{\frac{4x^4 - 11x^2 + 9}{|x|}}\right) = -f(x) \] Thus, \( f(-x) = -f(x) \), which proves that \( f(x) \) is an odd function.