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

Using the definition of the absolute value function:

The given function:

Becomes the piece-wise continuous function:

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

Therefore, this function does not have an inverse.

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To find the inverse of ( f(x) = |x-2| ), we swap the roles of ( x ) and ( y ) and then solve for ( y ):

( y = |x - 2| )

Swap ( x ) and ( y ):

( x = |y - 2| )

Solve for ( y ):

- If ( y - 2 \geq 0 ), then ( y - 2 = x ) → ( y = x + 2 )
- If ( y - 2 < 0 ), then ( -(y - 2) = x ) → ( y = 2 - x )

So, the inverse function is:

( f^{-1}(x) = \begin{cases} x + 2, & \text{if } x \geq 2 \ 2 - x, & \text{if } x < 2 \end{cases} )

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To find the inverse of the function ( f(x) = |x-2| ), follow these steps:

- Replace ( f(x) ) with ( y ): ( y = |x-2| ).
- Swap the variables ( x ) and ( y ): ( x = |y-2| ).
- Solve for ( y ).
- When ( y-2 \geq 0 ), ( |y-2| = y-2 ).
- When ( y-2 < 0 ), ( |y-2| = -(y-2) ).

- Split into two cases:
- Case 1: ( y-2 \geq 0 ) (or ( y \geq 2 )): ( x = y-2 ) becomes ( y = x+2 ).
- Case 2: ( y-2 < 0 ) (or ( y < 2 )): ( x = -(y-2) ) becomes ( y = -x+2 ).

- Combine both cases: [ y = \begin{cases} x+2, & \text{if } y \geq 2 \ -x+2, & \text{if } y < 2 \end{cases} ]

Therefore, the inverse of ( f(x) = |x-2| ) is: [ f^{-1}(x) = \begin{cases} x+2, & \text{if } x \geq 2 \ -x+2, & \text{if } x < 2 \end{cases} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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