Recently, I've become interested in hyperoperations. I wondered what the equation y=x tetrated by x (x[4]x), which is the same as x pentated by 2 (x[5]2), would look like on a graph. To do this, though, would mean to tetrate a number by a non-integer value, and everywhere I've looked has just come up by saying that there just isn't any known way to compute this.
For a moment I tried to figure out a way but the properties of previous hyperoperations just don't carry over to tetration.
It's weird to me that this can be done with addition, multiplication, and exponentation but no further. Maybe we're just not used to looking at it in a certain way, I don't know, I'm not experienced in maths, I'm just curious.
Mainly, I just want to know if there is any/what research is being done on tetrating a number by a non-whole number/fraction. Or if there is any proposed way to calculate an answer to one of those problems.
Thanks in advance.