Given the function:
f(x) = -sqrt(4-x^2)
We need to find the inverse.
Let us assume that y= - sqrt(4-x^2)
Let us square both sides.
==> y^2 = (4-x^2)
Now we will rewrite:
==> x^2 = 4- y^2
==> x = +- sqrt(4-y^2)
Now we will replace y and x.
==> y= +-sqrt( 4-x^2)
But since -2 =< x =< 0 , then we will not consider sqrt(4-x^2) as an inverse.
Then the inverse function is:
f^-1 ( x) = - sqrt(4-x^2) when -2 =< x =< 0
We have to find the inverse function of f(x) = -sqrt (4 – x^2)
Now as -2<= x < = 0, we have -2<= y < = 0, also.
Let y = f(x) = -sqrt (4 - x^2)
=> -y = sqrt (4 – x^2)
=> y^2 = 4 – x^2
=> x^2 = 4 – y^2
=> x = sqrt (4 – y^2) or x = -sqrt (4 – y^2)
We have to select x = -sqrt (4 – y^2), due to the condition imposed earlier.
Interchange y and x
=> y = f^-1(x) = -sqrt (4 – x^2)
Therefore we see that the inverse function f^-1(x) = -sqrt (4 - x^2) for -2<= x < = 0, is the same function.