Inverse Functions

If $f\left(x\right)={\left[4-{\left(x-7\right)}^3\right]}^{\frac{1}{5}}$, find the inverse of the function $f^{-1}\left(x\right)$.

If $y=g\left(x\right)=\frac{ax+b}{cx-a}$, then prove that $g\left(y\right)=x$.

If $\overrightarrow{r}:\mathrm{W}⟶\mathrm{W}$ defined by $f\left(x\right)=x-1$, if $x$ is odd and $f\left(x\right)=x+1$, if $x$ is even. Show that $f$ is invertible. Find the inverse of $f$, where $\mathrm{W}$ is the set of all whole numbers.

Show that the function $f$ in $\mathrm{A}=\mathrm{R}-\left\{\frac{2}{3}\right\}$ defined as $f\left(x\right)=\frac{4x+3}{6x-4}$ is one-one and onto. Hence, find $f^{-1}$.

Let $\mathrm{A}=\mathrm{R}-\left\{-\frac{5}{2}\right\}$ and $\mathrm{B}=\mathrm{R}-\left\{\frac{3}{2}\right\}$. Consider the function $f:\mathrm{A}\to \mathrm{B}$ defined by $f\left(x\right)=\frac{3x+2}{2x+5}$. Show that $f$ is one-one and onto function. find the inverse of $f$ and hence find $x$ such that $f^{-1}\left(x\right)=2$.

Let $\mathrm{A}=\mathrm{R}-\left\{-\frac{5}{2}\right\}$ and $\mathrm{B}=\mathrm{R}-\left\{\frac{3}{2}\right\}$. Consider the function $f:\mathrm{A}\to \mathrm{B}$ defined by $f\left(x\right)=\frac{3x+2}{2x+5}$. Show that $f$ is one-one and onto function. find the inverse of $f$ and hence, find $f^{-1}\left(0\right)$.

Let $\mathrm{A}=\mathrm{R}-\left\{-\frac{2}{3}\right\}$ and $\mathrm{B}=\mathrm{R}-\left\{\frac{2}{3}\right\}$ consider the function $f:\mathrm{A}⟶\mathrm{B}$ defined by $f\left(x\right)=\frac{2x+3}{3x+2}$ show that $f$ is one-one and onto function. Hence, find $f^{-1}$.

Consider $f:{\mathrm{R}}^{+}⟶[-9,\infty )$ define by $f\left(x\right)=5x^2+6x-9$. Show that $f$ is invertible with $f^{-1}\left(y\right)=\frac{\sqrt{5y+54}-3}{5}$.

Let ${\mathrm{R}}^{+}$ be the set of all positive real numbers and $f:{\mathrm{R}}^{+}⟶[4,\infty )$ define by $f\left(x\right)=x^2+4$. Show that. $f$ is invertible and find $f^{-1}$.

Find the inverse function of $f\left(x\right)=\frac{x-1}{x+1}, x\ne -1$, and verify that $fo{f}^{-1}$ is an identity function for all $x\in \mathrm{R}$.

Let $\mathrm{A}=\mathrm{R}-\left\{2\right\}$ and $\mathrm{B}=\mathrm{R}-\left\{1\right\}$. Consider the function $f:\mathrm{A}⟶\mathrm{B}$ defined by $f\left(x\right)=\frac{x-1}{x-2}$. Show that $f$ is one-one and onto function. Hence, find $f^{-1}$.

Let $\mathrm{A}=\mathrm{R}-\{-1\}$ and $\mathrm{B}=\mathrm{R}-\left\{1\right\}$. Consider the function $f:\mathrm{A}⟶\mathrm{B}$ defined by $f\left(x\right)=\frac{x-3}{x+1}$. Show that $f$ is one-one and onto function. Hence, find $f^{-1}$.

Let $\mathrm{A}=\mathrm{R}-\left\{3\right\}$ and $\mathrm{B}=\mathrm{R}-\left\{1\right\}$. Consider the function $f:\mathrm{A}⟶\mathrm{B}$ defined by $f\left(x\right)=\frac{x-2}{x-3}$. Show that $f$ is one-one and onto function. Hence, find $f^{-1}$.

If $f:\mathrm{R}⟶\mathrm{R}$ is defined by $f\left(x\right)=x^2+2$, find $f^{-1}\left(6\right), f^{-1}\left(11\right), f^{-1}(-2)$
 

If $f:\mathrm{R}⟶\mathrm{R}$ is defined by $f\left(x\right)=3x+5$, find $f^{-1}\left(8\right), f^{-1}\left(5\right), f^{-1}(-1)$.

Let $\mathrm{A}=\{2,3,4,5,7\}$ and $\mathrm{B}=\{3,4,6,8\}$ and let $f:\mathrm{A}⟶\mathrm{B}$ given by $f\left(2\right)=3, f\left(3\right)=4, f\left(4\right)=8, f\left(5\right)=3, f\left(7\right)=8$. Find the inverse images of all elements of $\mathrm{B}$.

Define 'Inverse Function'. Give one example.

Draw the graph of inverse of the function $f\left(x\right)=3x^2$. defined by the relation $R=\left\{\left(0,0\right),\left(1,3\right),\left(2,12\right),\left(3,27\right),\left(4,48\right)\right\}$

Draw the graph of inverse of the function $f\left(x\right)=2x^2$. defined by the relation $R=\left\{\left(0,0\right),\left(1,2\right),\left(2,8\right),\left(3,18\right),\left(4,32\right)\right\}$

Draw the graph of inverse of the function $f\left(x\right)=4x^2$. defined by the relation $R=\left\{\left(0,0\right),\left(1,4\right),\left(2,16\right),\left(3,36\right),\left(4,64\right)\right\}$