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

1 Answer
Mar 30, 2017

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.