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 ^-1x,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 ^-1x,

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 ^-1x 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 ^-1x

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

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

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 ^-1x, under certain conditions. Here ,

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

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 ^-1x is meaningful

Thus, Domain: x ε R

Range: y ε (0,π)

(5). If sec y =x, then y=sec ^-1x, 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 ^-1x, 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

Trigonometry

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