The inverse of $y = 5^{\log x}$ is

Let $f: R - \{\frac{\alpha}{6}\} \rightarrow R$ be defined by $f(x) = \frac{5x + 3}{6x - \alpha}$. Then the value of $\alpha$ for which $(f \circ f)(x) = x$,for all $x \in R - \{\frac{\alpha}{6}\}$,is:

Let $A = \{1, 2, 3\}$ and $B = \{1, 3, 5\}$. If a relation $R$ is defined from $A$ to $B$ as $R = \{(1, 3), (2, 5), (3, 3)\}$,then find $R^{-1}$.

Let $f: X \rightarrow Y$ be an invertible function. Show that the inverse of $f^{-1}$ is $f$,i.e.,$\left(f^{-1}\right)^{-1}=f$.

The inverse function of $f(x) = \frac{8^{2x} - 8^{-2x}}{8^{2x} + 8^{-2x}}, x \in (-1, 1),$ is

Let $Y = \{n^{2} : n \in N\} \subset N$. Consider $f: N \rightarrow Y$ as $f(n) = n^{2}$. Show that $f$ is invertible. Find the inverse of $f$.