How do you find the inverse of an equation?

Generally, to find the inverse of an equation, replace every x with y and replace every y (otherwise written f(x) ) with an x. Then it's "good form" to get the equation into y= form.

For an equation involving only two variables, the inverse can be found by swapping the x and y variables. Then, solve for y. If the equation does not define y as a function of x, the function f does not have an inverse. In order to start talking about an inverse, be sure first, that the given equation defines y as a function of x. Not every graph in the rectangular system is the graph of a function. For example, if you have an equation:

x^2/4 + y^2/9 = 1

it's wrong to say the inverse will be:

y^2/4 + x^2/9 = 1. Both of the above equations are ellipses. The original equation is an ellipse with the major axis (the long axis) on the y-axis, while the other has the major axis on the x-axis. Both of them do not represent a function, because if you solve for y, you'll see that two values of y can be obtained for a given x. Please note that if you are talking about functions, then not every function has an inverse, as a function must be one-to-one in order to have an inverse. A function must pass the "horizontal line test", which states that the graph of a function must never intersect with a horizontal line more than once, anywhere on it's domain. Inverse functions have some special properties:

1) The graph of an inverse function is the reflection of the original function reflected across the line y = x.

2) A function and it's inverse cancel each other out through functional composition.