Finding the inverse of a modulus function

In summary, the given function y=|x-4| does not have an inverse due to not being a one-to-one function. The function needs to be separated at x= -4 in order to have an inverse. For x≥-4, the inverse is y=x+4 and for x<-4, the inverse is y=-x-4. Both inverses are defined only for x≥0.
  • #1
rreedde
3
0

Homework Statement


Find the inverse of
y=|x-4|


Homework Equations


-


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?
 
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  • #2
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 [itex]x\ge -4[/itex], [itex]x+ 4\ge 0[/itex] so y= x+4. The inverse of that is, of course, y= x-4 defined only for [itex]x\ge 0[/itex]. 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 [itex]x\ge 0[/itex].
 
  • #3


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.
 

Related to Finding the inverse of a modulus function

1. What is a modulus function?

A modulus function, also known as the remainder function, is a mathematical function that returns the remainder after dividing one number by another. It is denoted by the symbol "%".

2. What is the inverse of a modulus function?

The inverse of a modulus function is a function that "undoes" the modulus operation. In other words, it takes the remainder and returns the original number that was divided.

3. How do you find the inverse of a modulus function?

To find the inverse of a modulus function, you can use the extended Euclidean algorithm. This involves finding the greatest common divisor between the original number and the modulus, and then using that to solve a linear congruence equation.

4. Can a modulus function have more than one inverse?

No, a modulus function can only have one inverse. This is because the modulus function is not a one-to-one function, meaning that different inputs can produce the same output.

5. In what situations is finding the inverse of a modulus function useful?

Finding the inverse of a modulus function is useful in various situations, such as in cryptography and error correction codes. It is also used in solving modular arithmetic equations and in finding solutions to linear congruences.

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