Finding the inverse of a modulus function

[rreedde]

Homework Statement

Find the inverse of
y=|x-4|

Homework Equations

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The Attempt at a Solution

i tried y+4=|x|
replacing y with x,
x+4= |y|
and I am quite stuck because of the modulus sign.
do i go on with x+4=y or -x-4=y?

 

[HallsofIvy]

There is a fundamental problem here: y= |x+ 4| is NOT 'one to one' and so does NOT have an inverse! In order to have 'inverses', we would neet to separate the function at x= -4. For $x\ge -4$, $x+ 4\ge 0$ so y= x+4. The inverse of that is, of course, y= x-4 defined only for $x\ge 0$. For x< -4, x+ 4< 0 so |x+4|= -(x+4)= -x- 4 so y= -x- 4. The inverse of that is y= -x- 4 again. And that, also, is defined only for $x\ge 0$.

 

[Architeuthis Dux]

I would approach this problem by first understanding the concept of an inverse function. The inverse of a function is a new function that "undoes" the original function. In other words, if we apply the original function and then the inverse function, we should get back to the original input.

In the case of a modulus function, we can think of it as a function that takes the absolute value of its input. So, the inverse function would need to "undo" this absolute value operation.

To find the inverse of y=|x-4|, we can start by setting y equal to a new variable, let's say u. This gives us u=|x-4|. Now, we can approach this problem algebraically by considering two cases: when x-4 is positive and when it is negative.

Case 1: x-4 is positive
In this case, the modulus function has no effect, so we can rewrite the equation as u=x-4.

Case 2: x-4 is negative
In this case, the modulus function would make it positive, so we can rewrite the equation as u=-(x-4).

Now, we can solve for x in each case:
Case 1: u=x-4
x=u+4

Case 2: u=-(x-4)
x=4-u

So, we have two possible solutions for x depending on the value of u. To combine these two solutions into one inverse function, we can use a piecewise function:

f^-1(u) = {u+4, if u ≥ 0
{4-u, if u < 0

This inverse function will "undo" the absolute value operation and give us back the original input. In other words, if we plug in the output of this inverse function into the original modulus function, we will get back the input value.