Relations and Functions — Practice Questions

For a suitably chosen real constant $a$, let a function, $f: \mathbf{R} - \{-\mathbf{a}\} \to \mathbf{R}$ be defined by $f(x) = \frac{a - x}{a + x}$. Further supposed that for any real number $x \neq -a$, and $f(x) \neq -a$, $(\mathrm{fof})(x) = x$. Then $f\left(-\frac{1}{2}\right)$ is equal to:

If the function $f \colon -\infty, -1 \to a, b$ defined by $fx = e^{x^3 - 3x + 1}$ is one-one and onto, then the distance of the point $P2b + 4$, $a + 2$ from the line $x + e^{-3}y = 4$ is:

Let $A = 1,2,3,\ldots 100$. Let $R$ be a relation on $A$ defined by $x, y \in R$ if and only if $2x = 3y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $n$. Then the minimum value of $n$ is ______.

If $fx = \frac{4x + 3}{6x - 4}$, $x \neq \frac{2}{3}$ and $(fof) \quad (x) = g(x)$, where $g: R - \frac{2}{3} \to R - \frac{2}{3}$, then $(gogog) \quad (4)$ is equal to