and 
be two functions from 
into , defined by 
and 
for all 
. Find f o and g o .Solution: $\begin{array}{ll} \mathrm{f}(\mathrm{x})=|\mathrm{x}|+\mathrm{x} \quad & \text { (given }) \\ \mathrm{g}(\mathrm{x})=|\mathrm{x}|-\mathrm{x} \quad & \text { (given) } \end{array}$...
and 
If 
, show that is one-one and onto. Hence, find 
Solution: $\mathrm{f}(\mathrm{x})=\frac{x-1}{x-2} \quad \text { (as given) }$ One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}$ Therefore, $f(p)-f(q)$...
Show that f: 
range 
is invertible. Find Solution: $\mathrm{f}(\mathrm{x})=4 \mathrm{x} 2+12 \mathrm{x}+15 \quad \text { (as given) }$ $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto...
be the set of all positive real numbers. show that the function ![f: R_{4} \rightarrow[-5, \infty]: f(x)=\left(9 x^{2}+6 x-5\right)](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b71d72438933be58e65dbddebe7345c9_l3.png "Rendered by QuickLaTeX.com")
is invertible. Find .Solution: $f(x)=9 x_{2}+6 x-5 \text { (as given) }$ $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One function Suppose p,q be two...
on 
into itself, defined by 
is one-one and onto. Hence, find Solution: $\mathrm{f}(\mathrm{x})=\frac{4 x}{3 x+4} \quad$ (as given) One-One function Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}$ Therefore, $f(p)=f(q)$...
on 
, defined as 
is one-one and onto. Hence, find ,Solution: $\mathrm{f}(\mathrm{x})=\frac{4 x+3}{6 x-4} \quad \text { (given) }$ $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One...
, show that (f 
for all 
.Solution: $\mathrm{f}(\mathrm{x})=\frac{4 x+3}{6 x-4}, \mathrm{x} \neq \frac{2}{3} \text { (given) }$ We need to Show: fof $(x)=x$ for all $x \neq \frac{2}{3}$ It is known that fof $(x)=f(f(x))$...
Solution: $\mathrm{f}(\mathrm{x})=\frac{1}{2}(3 \mathrm{x}+1) \quad$ (given) $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One...
.Solution: $\mathrm{f}(\mathrm{x})=3 \mathrm{x}-4 \quad$ (as given) $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One function...
.Solution: $\mathrm{f}(\mathrm{x})=2 \mathrm{x}+3$ (as given) $\mathrm{f}(\mathrm{x})$ is invertible if $\mathrm{f}(\mathrm{x})$ is a bijection (i.e one-one onto function) One-One function Suppose...
.Solution: A function is invertible if it is a bijection. (i.e. One-One Onto function) One-One function $\mathrm{f}=\{(2,7),(3,9),(4,11),(5,13)\}$ It is observed that different elements of A have...