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

Let $f : R \rightarrow  R\ f(x) = x^3 -3x^2 + 3x\ -2$ , then $f^{-1}(x)$ is given by

Consider $f: R _{+} \rightarrow[4, \infty)$ given by $f(x)=x^{2}+4 .$ Show that $f$ is invertible with the inverse $f^{-1}$ of $f$ given by $f^{-1}(y)=\sqrt{y-4},$ where $R ^{+}$ is the set of all non-negative real numbers.

If the function $f:[1,\;\infty ) \to [1,\;\infty )$ is defined by $f(x) = {2^{x(x – 1)}},$ then ${f^{ – 1}} (x)$ is

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