Question

How do you find the inverse of `f(x) = |x-2|`?

Answer 1

Using the definition of the absolute value function:

`|a| = {(a; a >=0), (-a; a < 0):}`

The given function:

`f(x) = |x-2|`

Becomes the piece-wise continuous function:

`f(x) = {(x - 2; x >=2), (-(x-2); x < 2):}`

The formal way to find an inverse of a function is:

Substitute `f^-1(x)` for x into `f(x)`:

`f(f^-1(x)) = {(f^-1(x) - 2; x >=2), (-(f^-1(x)-2); x < 2):}`

Invoke the property `f(f^-1(x))`

`x = {(f^-1(x) - 2; f^-1(x) >=2), (-(f^-1(x)-2); f^-1(x) < 2):}`

And then solve for `f^-1(x)`

But we cannot do this, because this will give us two equations for `f^-1(x)` which indicates that it is not a function.

Therefore, this function does not have an inverse.