Ucale

# Inverse Function

If a function is one to one and onto from A to B, then function g which associates each elementÂ Â $\displaystyle y\quad \epsilon \quad B$ to one and only one elementÂ $\displaystyle x\quad \epsilon \quad A$

such that y=f(x) then g is called the inverse function of f, denoted by x=g(y)

Usually we denote g = fÂ -1 {Read as f inverse }Â x=fÂ -1(y)

## Domain and Range of Inverse Functions

(1) If sin y=x, then y=sinÂ -1Â x,under certain conditionÂ

$\displaystyle -1\le \sin { y } \le 1$ but sin y =x

$\displaystyle \therefore -1\le x\le 1$

againÂ $\displaystyle \sin { y } =-1\Rightarrow y=-\frac { \pi }{ 2 } \\ \sin { y } =1\Rightarrow y=\frac { \pi }{ 2 }$ and keeping in mind numerically smallest angles or real numbers.

$\displaystyle \therefore \frac { \pi }{ 2 } \le y\le \frac { \pi }{ 2 }$

These restrictions on the values of x and y provide us with the domain and range for the function y=sinÂ -1Â x,

Domain:Â $\displaystyle x\quad \epsilon \quad \left[ -1,1 \right]$

Range:Â $\displaystyle y\quad \epsilon \quad \left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right]$

(2). Let cos y=x thenÂ  y=cosÂ -1Â x under certain conditions

$\displaystyle -1\le \cos { y } \le 1\\ \Rightarrow 1\le x\le 1\\ \cos { y } =-1\Rightarrow y=\pi \\ \cos { y } =1\Rightarrow y=0$

$\displaystyle \therefore 0\le y\le \pi$ as cos x is a decreasing function in [0,Ï€];

henceÂ $\displaystyle \cos { \pi } \le \cos { y } \le \cos { 0 }$

These restrictions on the values of x and y provide us the domain and range for the functionÂ  y=cosÂ -1Â x

(3).Â If tan y= x then y=tanÂ -1Â x, under certain conditions. Here ,

$\displaystyle \tan { y } \epsilon \quad R\\ \Rightarrow x\quad \epsilon \quad R\\ -\infty <\tan { y } <\infty \\ \Rightarrow -\frac { \pi }{ 2 }

Thus, domain xÂ  Îµ R

RangeÂ $\displaystyle y\quad \epsilon \quad \left( -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right)$

(4). If cot y =x , then y=cotÂ -1Â x, under certain conditions. Here ,

$\displaystyle \cot { y } \epsilon \quad R\\ \Rightarrow x\quad \epsilon \quad R\\ -\infty <\cot { y } <\infty \\ \Rightarrow 0

These conditions on x and y make the function, cot y = x one-one and onto so that the inverse function exists.

i.e. y=Â cotÂ -1Â x is meaningful

Thus, Domain:Â  xÂ  Îµ R

Range: y Îµ (0,Ï€)

(5). If sec y =x, then y=secÂ -1Â x, under certain conditions. Here ,

$\displaystyle \left| x \right| \ge 1\\ 0\le y\le \pi ,\quad y\neq \frac { \pi }{ 2 }$

Domain:Â  xÂ  Îµ R

Range: y Îµ [0,Ï€] -[Ï€/2]

(6). IfÂ  cosec y =x thenÂ y=cosecÂ -1Â x, where ,Â

$\displaystyle \left| x \right| \ge 1\\ 0\le y\le \pi ,\quad y\neq \frac { \pi }{ 2 }$

domain xÂ  Îµ R

RangeÂ $\displaystyle y\quad \epsilon \quad \left( -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right)$

February 22, 2019
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