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>> Inverse Function Part-1
Let f : X → Y be a function defined by y = f(x) such that f is both one - one and onto. Then there exists a unique function g : Y → X such that for each y ε Y,
g(y) = x <=> y = f(x). The function g so defined is called the inverse of f.
Further, if g is the inverse of f, then f is the inverse of g and the two functions f and g are said to be the inverses of each other. For the inverse of a function to exists, the function must be on-one and onto.
Method to Find Inverse of a Function
If f-1 be the inverse of f, then fof-1 = f-1 of = I, where I is an identity function.
fof-1 = I => (fof-1(x)) = I (x) = x.
Apply the formula of f on f-1 (x), we will get an equation in f-1 (x) and x.
Solve it to get f-1 (x).
Note : A function and its inverse are always symmetric with respect to the line y = x.
Let f : R → R defined by f(x) = (ex-e-x)/2 . Find f-1 (x).
Solution: We have f(f-1(x)) = x
=> (ef-1(x) - e-f-1(x))/2 = x
=> e2f-1(x) - 2xef-1(x) -1 = 0
=> ef-1(x) = x + √(x2 +1).
But negative sign is not possible because L.H.S. is always positive.
Thus ef-1(x) = x + √(x2 +1) . Hence, f-1(x) = log(x + √(x2 +1)) .
We give below some standard functions along with their inverse functions: