Question

Find the inverse of the function on the given domain.f(x)=(x−1)^2 , [1,∞)

Answer 1

We have the function:

`f(x)=(x-1)^2`

defined for the domain [1,∞).

We have to find its inverse.

We can use the fact that if f(x) and g(x) are inverse functions, the following condition will be true:

`f(g(x))=x`

We can then use it to find g(x):

`\begin{gathered} f(g(x))=x \\ (g-1)^2=x \\ (g-1)=√(x) \\ g(x)=√(x)+1 \end{gathered}`

The domain of f(x) will be the range of its inverse function g(x). Also, the range of f(x) will become the domain of its inverse g(x).

Also, given the domain of f(x), we know that the square root is positive and not negative, as the range would be different.

We can graph the functions as:

Answer: the inverse function is f^(-1)(x) = √x+1.