Amit M Agarwal Solutions for Chapter: Relations and Functions, Exercise 5: Target Exercises

Let $f:R\to R$ be given $f\left(x\right)={\left(x+1\right)}^2-1, x\ge -1$. Then, the set of values of $x$ for which $f\left(x\right)=f^{-1}\left(x\right)$, is given by

Let $f:R\to R$ be a function defined by $f\left(x\right)=x-\left[x\right]$, where $\left[ \right]$ denotes the greatest integer function, then $f^{-1}\left(x\right)$ is

The inverse of the function $y=\frac{{10}^x-{10}^{-x}}{{10}^x+{10}^{-x}}+1$ is

The inverse of the function $f\left(x\right)=\frac{e^x-e^{-x}}{e^x+e^{-x}}+2$ is given by

Let $f:R\to R,g:R\to R$be two functions given by $f\left(x\right)=2x-3,g\left(x\right)=x^3+5$. Then ${\left(fog\right)}^{-1}\left(x\right)$ is equal to

Let $f:\left[2,\infty \right)\to X$ be defined by $f\left(x\right)=4x-x^2$, Then, $f$ is invertible, if $X$ is equal to

If $f:R\to R$ is given by $f\left(x\right)=x^3+3$, then $f^{-1}\left(x\right)$ is equal to

If $f:\left[1,\infty \right)\to \left[2,\infty \right)$ is given by $f\left(x\right)=x+\frac{1}{x}$, then $f^{-1}\left(x\right)$ is equal to