**Inverse Function**

If a function is one to one and onto from A to B, then function g which associates each elementÂ Â to one and only one elementÂ

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 **

againÂ and keeping in mind numerically smallest angles or real numbers.

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

Range:Â

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

as cos x is a decreasing function in [0,Ï€];

henceÂ

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 ,**

Thus, domain xÂ Îµ R

RangeÂ

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

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 ,**

Domain:Â xÂ Îµ R

Range: y Îµ [0,Ï€] -[Ï€/2]

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

domain xÂ Îµ R

RangeÂ