Question:
show that f:-1,1]...R given by f(x) =x/x+2 is one one. find the inverse of the function f:[-1,1]...range (f)
Answer:
Given, f(x) = x/(x + 2)
Let x1 , x2 ∈ R
Now, f(x1 ) = f(x2 )
=> x1 /(x1 + 2) = x2 /(x2 + 2)
=> x1 * (x2 + 2) = x2 * (x1 + 2)
=> x1 * x2 + 2x1 = x2 * x1 + 2x2
=> 2x1 = 2x2
=>x1 = x2
Hence, f is one-one function.
Let y = x/(x + 2)
=> y(x + 2) = x
=> xy + 2y = x
=> 2y = x - xy
=> x(1- y) = 2y
=> x = 2y/(1 - y) ..................1
So, inverse of f(x) = 2x/(1 - x)
Now from equation 1,
x is not defined when y = 1
So, range (f) = R - 1
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