# How do you find the inverse of  y = | x - 3 |?

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Rhys Share
Dec 17, 2017

As scene as below

#### Explanation:

This is an interesting problem!

I am going to first sketch the function:

To how we can then say:

$y = x - 3 , x \ge 3$
$y = 3 - x , x \le 3$

We can now find the inverse of both of these individual graphs:

$y \left(x\right) = x - 3$
$\implies x = {y}^{-} 1 \left(x\right) - 3$
$\implies {y}^{-} 1 \left(x\right) = x + 3$

${y}_{1} \left(x\right) = 3 - x$
$\implies x = 3 - {y}_{1}^{-} 1 \left(x\right)$
$\implies {y}_{1}^{- 1} \left(x\right) = 3 - x$

Hence the inverse:

$y = 3 - x , x \ge 0$
$y = x + 3 , x \ge 0$

We can see that $x \ge 0$ with a sketch of the reflection in $y = x$:

$y = | x - 3 |$ can also be inversed via swappign the $x$ for the $y$:

$x = | y - 3 |$

Hope this helped you, or at least prompted you in the right direction!

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