If $g$ is the inverse of $f$ and $f^{\prime}(x)=\frac{1}{1+x^{2}}$,then $g^{\prime}(x)$ is equal to

Let $f: N \rightarrow R$ be a function defined as $f(x)=4x^{2}+12x+15$. Show that $f: N \rightarrow S$,where $S$ is the range of $f$,is invertible. Find the inverse of $f$.

Consider $f: \{1, 2, 3\} \rightarrow \{a, b, c\}$ and $g: \{a, b, c\} \rightarrow \{\text{apple, ball, cat}\}$ defined as $f(1)=a, f(2)=b, f(3)=c$ and $g(a)=\text{apple}, g(b)=\text{ball}, g(c)=\text{cat}$. Show that $f, g$ and $g \circ f$ are invertible. Find $f^{-1}, g^{-1}$ and $(g \circ f)^{-1}$ and show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

If $f(x) = 3x - 5$,then ${f^{ - 1}}(x)$ is:

If $f(x) = \frac{x}{1 + x}$,then ${f^{-1}}(x)$ is equal to

Let $f(x)=(x+1)^2-1, x \geqslant-1$,then the set $\{x : f(x)=f^{-1}(x)\}$ is