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For the general function f(x) = sqrt (ax + b), the inverse can be computed as follows.
let y = sqrt (ax + b)
=> y^2 = ax + b
=> ax = y^2 - b
=> x = (y^2 - b)/a
interchange x and y
=> y = (x^2 - b)/a
The inverse of f(x) = sqrt (ax + b) is (x^2 - b)/a
For polynomials of higher order under the square root sign, deriving the inverse would become considerably difficult.
First, we'll note y= sqrt(ax+b).
Now we'll change x by y:
We'll raise to square both sides to eliminate the square root:
x^2 = ay + b
We'll isolate y to the left side:
-ay = b - x^2
We'll divide by -a
y = x^2/a - b/a
The inverse function is:
f^-1(x) = x^2/a - b/a
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