Let $S = \{a, b, c\}$ and $T = \{1, 2, 3\}$. Find $F^{-1}$ of the following function $F$ from $S$ to $T$,if it exists: $F = \{(a, 3), (b, 2), (c, 1)\}$.

If the functions $f$ and $g$ are defined by $f(x) = 3x - 4$ and $g(x) = 2 + 3x$ for $x \in R$,then $g^{-1}(f^{-1}(5))$ is equal to

Let $f: R-\{2\} \rightarrow R-\{1\}$ defined by $f(x)=\frac{x-3}{x-2}$ and $g: R \rightarrow R$ defined by $g(x)=3x-2$. Then,the sum of all values of $x$ for which $f^{-1}(x)+g^{-1}(x)=\frac{19}{6}$ 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 $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ and $g: R \rightarrow (0, 4)$ be functions defined by $f(x) = \log_e(x^2 + 2x + 4)$ and $g(x) = \frac{4}{1 + e^{-2x}}$. Define the composite function $h(x) = (f \circ g^{-1})(x)$,where $g^{-1}$ is the inverse of the function $g$. Then the value of the derivative of the composite function $h(x)$ at $x = 2$ is:

Which of the following functions is the inverse of itself?