Ucale

# Inverse Function

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

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

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

## Domain and Range of Inverse Functions

(1) If sin y=x, then y=sin -1 x,under certain condition

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

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

again $\displaystyle \sin { y } =-1\Rightarrow y=-\frac { \pi }{ 2 } \\ \sin { y } =1\Rightarrow y=\frac { \pi }{ 2 }$ 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=sin -1 x,

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

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

(2). Let cos y=x then  y=cos -1 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,π];

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

These restrictions on the values of x and y provide us the domain and range for the function  y=cos -1 x

(3). If tan y= x then y=tan -1 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 x  ε R

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

(4). If cot y =x , then y=cot -1 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= cot -1 x is meaningful

Thus, Domain:  x  ε R

Range: y ε (0,π)

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

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

Domain:  x  ε R

Range: y ε [0,π] -[π/2]

(6). If  cosec y =x then y=cosec -1 x, where ,

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

domain x  ε R

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

February 22, 2019
Which class you are presently in?
Choose an option. You can change your option at any time.
You will be solving questions and growing your critical thinking skills.