Inverse Function of Greatest Integer Function in a Given Domain

[Cookiey]

Homework Statement

If f:(2,4)-->(1,3) where f(x)=x-[x/2] (where[.] denotes the greatest integer function), then find the inverse function of f(x).

Homework Equations

(None I believe.)

The Attempt at a Solution

I know that for a function to be invertible, it must be both one-one and onto.

I've checked if the function is one-one by drawing a graph in the given domain. It comes out to be a straight line, and so a line drawn parallel to the x-axis will intersect it at only one place; meaning it is one-one. From the same graph, it's clear that the range equals the codomain, making it onto.

So, the function is definitely invertible.

I don't really know how to proceed in inverting the function. I've tried splitting 'x' into it's integer and fractional parts so I can take [x] common, but that leaves out {x}.

So, I took a look at the three options, namely:
a)not defined
b)x+1
c)x-1

(a) is definitely out, as the function is invertible in the given domain, as proved earlier.

Looking at (b) and (c), I noticed that in the given domain, [x/2] always evaluates to 1, and therefore the answer would be (b), which is the correct answer.

But is there a 'proper' way to do this? A way to work with the greatest integer function in general in a given domain/range?

Thank you for taking the time to read and help me!

 

[Mark44]

Homework Statement

If f:(2,4)-->(1,3) where f(x)=x-[x/2] (where[.] denotes the greatest integer function), then find the inverse function of f(x).

Homework Equations

(None I believe.)

The Attempt at a Solution

I know that for a function to be invertible, it must be both one-one and onto.

I've checked if the function is one-one by drawing a graph in the given domain. It comes out to be a straight line, and so a line drawn parallel to the x-axis will intersect it at only one place; meaning it is one-one. From the same graph, it's clear that the range equals the codomain, making it onto.

So, the function is definitely invertible.

I don't really know how to proceed in inverting the function. I've tried splitting 'x' into it's integer and fractional parts so I can take [x] common, but that leaves out {x}.

So, I took a look at the three options, namely:
a)not defined
b)x+1
c)x-1

(a) is definitely out, as the function is invertible in the given domain, as proved earlier.

Looking at (b) and (c), I noticed that in the given domain, [x/2] always evaluates to 1, and therefore the answer would be (b), which is the correct answer.

But is there a 'proper' way to do this? A way to work with the greatest integer function in general in a given domain/range?

Thank you for taking the time to read and help me!

I don't know that there is a "proper" way to do this. The usual algebraic tricks don't apply because of the greatest integer function in this problem. What you did is more than likely what you were supposed to do.

 

[Cookiey]

I don't know that there is a "proper" way to do this. The usual algebraic tricks don't apply because of the greatest integer function in this problem. What you did is more than likely what you were supposed to do.

I see! Thank you for replying! :)

 

[HallsofIvy]

Yes, it is true that for $2\le x\le 4$, $1\le x/2\le 2$ so that $[x]= 1$. The problem, then, is simply to find the inverse function to f(x)= x- 1.

The simplest way to find the inverse function of that is to say "f(x) simply subtracts one from x- the opposite of that is to add 1. $$f^{-1}(x)= x+ 1$$". A more "algorithmic" method would be two write f(x)= y= x- 1 so the inverse function is given by x= y- 1. Solve for y by adding 1 to both sides and you get, again, $$f^{1}(x)= y= x+ 1$$.