How to we do the inverse of y=(x-1)^2 ?

[cmab]

How to we do the inverse of y=(x-1)^2 ?

Would it be x = sqrt(y) +1 ?

 

[Jameson]

Let's see.

$$y = (x-1)^2$$

Now switching x and y we get

$$x = (y-1)^2$$
$$\sqrt{x} = y - 1$$
$$y = \sqrt{x} + 1$$

When you find inverses, you usually want to put the inverse in terms of the given variable if possible. Sometimes you'll see, it is quite impossible.

Example: Find the inverse of $$y = x^3-x$$

Jameson

 

[cmab]

but in my problem, it must have respect to y...

How about the reciprocal of x=2, would it be y=2 ? Just swapping the variable.

 

[dimitri151]

The issue with finding the inverse of x=2 is that x=2 isn't a function. A function is an ordered tuple from one set to another. x=2 only refers to an one element of one set. Furthermore, if you want your function to have an inverse the rule has to have other requirements. It has to be bijective, which means it has to be both injective and surjective. Hence $$y = x^3-x$$
has no inverse since solving for x gives multiple functions.