1. For the function f(x)=-4(sqrt x-1), find the inverse function and x.
2. The function f(x)=(x-1)^2 - 4 is not one-to-one. If you restrict the doamin for f(x) to x≤1, what is its inverse function and the domain for the inverse?
+227 1. We have the equation \(f(x) = -4\sqrt{x}-1\)
At least, that's what I'll assume since you didn't add parenthesis around sqrt(). Anywho, to find the inverse, we just swap f(x) and x.
So, \(x=-4\sqrt{f(x)}-1\)
Solving for x gives:
\(f^{-1}(x) = (\frac{x+1}{-4})^2\)
That would be f^-1(x), the inverse function.
To prove that they are indeed inverse functions, we can use composition of functions. If they are inverses, we should obtain f(f^-1(x)) = x and f^-1(f(x)) = x.
\(f(f^{-1}(x)) = -4\sqrt{(\frac{x+1}{-4})^2}-1 = -4\frac{x+1}{-4}-1 = x+1-1 = x\)
\(f^{-1}(f(x)) = (\frac{-4\sqrt{x}-1+1}{-4})^2 = (\frac{-4\sqrt{x}}{-4})^2 = (\sqrt{x})^2 = x\)
Hence, these two functions are inverses.
2. This is quite similar to number one; you should be able to do it now.
