Let $f: R \to R$ be a function. Define $g: R \to R$ by $g(x) = |f(x)|$ for all $x$. Then $g$ is

$f(x) = \frac{x}{\ln x}$ and $g(x) = \frac{\ln x}{x}$. Identify the $CORRECT$ statement.

Let $P(x)$ be a polynomial with real coefficients such that $P(\sin^2 x) = P(\cos^2 x)$ for all $x \in [0, \pi/2)$. Consider the following statements:
$I.$ $P(x)$ is an even function.
$II.$ $P(x)$ can be expressed as a polynomial in $(2x - 1)^2$.
$III.$ $P(x)$ is a polynomial of even degree.
Then,

Let the function $f:[0,1] \rightarrow \mathbb{R}$ be defined by $f(x) = \frac{4^x}{4^x+2}$. Then the value of $f\left(\frac{1}{40}\right) + f\left(\frac{2}{40}\right) + f\left(\frac{3}{40}\right) + \dots + f\left(\frac{39}{40}\right) - f\left(\frac{1}{2}\right)$ is:

Let $f(x) = x^{12} - x^9 + x^4 - x + 1$. Which of the following is true?

Let the function $f(x) = x^2 + x + \sin x - \cos x + \log(1 + |x|)$ be defined over the interval $[0, 1]$. The odd extension of $f(x)$ to the interval $[-1, 1]$ is: