I Inverse Functions: x=f(y) and X=f^-1(y)

[DumpmeAdrenaline]

Consider the case of a real function f for which f inverse exists.

1) We we are not used to having the y-axis (vertical axis) to denote the independent variable which it does in x=f-1(y). We rotate the system through positive 90 degree and reflect about the vertical to change the sense of the axis. One further adjustment to make is to relabel the horizontal axis x and the vertical axis y (the names we give to the axes are immaterial).

If we accept the convention that the independent variable must be always plotted along the horizontal axis in the left right sense and the independent variable be called x. Then the first figure on the top left is the appropriate graph of y=f(x) and the bottom figure would be the appropriate graph for y=f^-1(x). Which figures are appropriate for these relations x=f(y) and x=f^-1(y)?

I think x=f(y) denotes the relation for the top left figure and x=f^-1(y) for the bottom figure because names to the input and output are irrelevant.

 

[PeroK]

Is there a question here?

 

[DumpmeAdrenaline]

Its stated in the study notes I am studying from that f(x)=y and f(y)=x are obtained from one another by merely reversing the roles of input and output. How is this possible? Arent both functions equal the difference between them two is what we choose to name as the input

 

[PeroK]

Its stated in the study notes I am studying from that f(x)=y and f(y)=x are obtained from one another by merely reversing the roles of input and output. How is this possible? Arent both functions equal the difference between them two is what we choose to name as the input

If you draw a graph of $y = f(x)$ and then rotate the graph by 90 degrees counter-clockwise, then that is the graph of $x = f^{-1}(y)$. It's not quite right, because the y-axis goes off to the left, rather then the right, so you also have to reflect the graph in what is now the vertical x-axis. That's just left/right convention.

Take as an example the graph of $y = x^3$. The point $(x=2, y=8)$ is on the graph. If we rotate the graph then that becomes the point $(y = 8, x=2)$ and we see that $x = y^{1/3}$. And, of course, the cube and cube root are inverse functions.

 

[Mark44]

If you draw a graph of $y = f(x)$ and then rotate the graph by 90 degrees counter-clockwise, then that is the graph of $x = f^{-1}(y)$. It's not quite right, because the y-axis goes off to the left, rather then the right, so you also have to reflect the graph in what is now the vertical x-axis. That's just left/right convention.

A possibly simpler way to think of this is that you can reflect the graph of $y = f(x)$ across the line y = x. That will give you the graph of $y = f^{-1}(x)$.

Assuming you start with a function that has an inverse, the graphs of $y = f(x)$ and $x = f^{-1}(y)$ are identical. The first equation gives you y as a function of x; the second equation gives you x as a (different; i.e., inverse) function of y.

Using the example given by @PeroK, the point (x=2, y = 8) lies on the graph of $y = x^3$. Exactly the same point lies on the graph of $x = y^{1/3}$.