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

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Answer 1 · 2017-03-30T17:53:39

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.

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