The other day in my Physics class we had some sample data that we wanted to linearize. The graph resembled a root curve. So to linearize it, we took the square root of all the x data and replotted the original Y and the new X. When we constructed the graph we called the original Y axis the original data type (Time). But on the X axis we labeled the data, square root of our original x data type (sqrt(length)). Basic linearizing a graph.
Then we got some other data and it looked like it was a polynomial of degree three. So with the help of the Stats teacher we linearized that graph by adjusting our x data by A(x)^3 + B(x)^2 + C(x) + D. We then graphed the original y with the new x and it became linear. However, what do we call the new x axis?
See picture below of an example of what we did. Thanks for all the help!
Do know about log-log graph and semi-log graph? In log-log graph, both x and y axes are labeled as x and y but log(x) vs log(y) are plotted. They label x and y on the axes but scales are changed from 0,1,2,3... to 1,10,100,1000,....
So in your case, if you label $x_1$ on the x-axis, then use normal scaling 0,1,2,3,... and write on side of the graph about $x_1 = f(x)$ and if you label $x$ on the x-axis, then change scaling accordingly and describe about it.
There're both advantages and disadvantages in both ways of graphing. In the first one, plotting is much easier but getting original value of $x$ can be very difficult (in your case it will be quite difficult to solve a cubic equation if you want a value of $x$). While in the second method, plotting will take little more time but it is easier to get the data from the graph. So usually the second method is used (as in case of log-log and semi-log graphs).
You just know that graph is a way to represent the data so make sure to not miss any important details and make it ambiguous for future use.