1. ## Vertical Triangle Wave

Is there a way to write a series of functions that describe the inverse of $\frac{2}{\pi}sin^{-1}sin(\pi x)$? I am aiming for a vertical triangle wave the travels up the y axis.

I think there may not be a series of functions that make this work, because any form of the relation in the vertical form violates the definition of a function...

Thanks!

2. Just take this plot and reflect it over the line y = x. You're correct in that there's no function that will do what you're asking, because it will be multi-valued. However, it is a relation, and you can plot it. The result should be something like this.

3. Originally Posted by masoug
Is there a way to write a series of functions that describe the inverse of $\frac{2}{\pi}sin^{-1}sin(\pi x)$? I am aiming for a vertical triangle wave the travels up the y axis.

I think there may not be a series of functions that make this work, because any form of the relation in the vertical form violates the definition of a function...

Thanks!
Maybe this helps (?):

$f_k(x)=\left\{\begin{array}{lccl}\frac12 x + 2k,&k\in\mathbb{Z},&x\in(-1,1] \\ -\frac12 x + 2k+1,&k\in\mathbb{Z},&x\in [-1,1) \end{array}\right.$

I've attached the graph of the given function and the family of straight lines forming the inverse of the function.

4. Originally Posted by Ackbeet
Just take this plot and reflect it over the line y = x. You're correct in that there's no function that will do what you're asking, because it will be multi-valued. However, it is a relation, and you can plot it. The result should be something like this.
So function plotters that I use: Function Grapher cannot graph this function...

Originally Posted by earboth
Maybe this helps (?):

$f_k(x)=\left\{\begin{array}{lccl}\frac12 x + 2k,&k\in\mathbb{Z},&x\in(-1,1] \\ -\frac12 x + 2k+1,&k\in\mathbb{Z},&x\in [-1,1) \end{array}\right.$

I've attached the graph of the given function and the family of straight lines forming the inverse of the function.
Hmm, so we need to use complex numbers to be able to graph the functions...

Thanks for the help anyway!

5. Originally Posted by masoug
So function plotters that I use: Function Grapher cannot graph this function...
Right. It's not a function, because it fails the vertical line test. You have to have software that will plot implicit functions, or multi-valued "functions".

Hmm, so we need to use complex numbers to be able to graph the functions...

Thanks for the help anyway!
I don't see where complex numbers come into it at all. Where are you seeing them?

6. Originally Posted by earboth

$f_k(x)=\left\{\begin{array}{lccl}\frac12 x + 2k,&k\in\mathbb{Z},&x\in(-1,1] \\ -\frac12 x + 2k+1,&k\in\mathbb{Z},&x\in [-1,1) \end{array}\right.$
$\mathbb{Z}$

I thought that meant complex number...

7. Originally Posted by masoug
$\mathbb{Z}$

I thought that meant complex number...
I'm sorry for the confusion: My apologies.

But in Germany $\mathbb{Z}$ denotes the set of all integers and $\mathbb{C}$ the set of complex numbers.

(But of course you are right that complex variables often are labeled with z)

8. Originally Posted by earboth
I'm sorry for the confusion: My apologies.

But in Germany $\mathbb{Z}$ denotes the set of all integers and $\mathbb{C}$ the set of complex numbers.
This is also the usual notation in the United States.