Determine whether the function has an inverse. Some rational functions do not have inverses. For a rational function to have an inverse, the function must be one-to-one, meaning that each element in the function's domain maps to exactly one element in the function's range. To determine whether a function is one-to-one, you must first graph the function by hand or by using a graphing calculator.
Perform the horizontal line test. This "test" states that if a horizontal line can be drawn anywhere on the function's graph and that line touches the function's graph more than once, then that function isn't one-to-one. Essentially, you are visually scanning the function to determine whether it is possible to draw a horizontal line that intersects the graph in more than one location, and if it is possible, then the function doesn't have an inverse. If your function fails the horizontal line test, then your answer is "no inverse." But, if it passes, as most tend to do, proceed to the next step.
Solve the equation for "x." This is equivalent to isolating "x" on one side of the equals sign with all other elements of the equation, the "y" term and any numbers, on the opposite side. To do this, you may need to add, subtract, multiply, divide, find square roots or any combination of these operations.
Consider the one-to-one function y = 2x -- 6. In order to isolate "x," first add "6" to both sides of the equation, producing y + 6 = 2x. Then divide both sides of the equation by 2, rendering (y + 6)/2 = x. Simplify on the left by first separating the terms into two distinct fractions: (y/2) + (6/2). Reduce any fractions to lowest terms, yielding (y/2) + 3 = x. If you wish, you may reverse the terms on each side of the equation, writing x = (y/2) + 3. You may also choose to write the "y" term as a decimal, in this case 0.5y.
Interchange "x" and "y." Replace the "x" with a "y" and the "y" with an "x." The example becomes y = (x/2) + 3 or y = 0.5x + 3.
Check your answer by graphing it on the same graph as the original function, along with the diagonal line y = x. The graph of a function's inverse is its reflection across the line y = x. The equation y = 0.5x + 3 reflects y = 2x -- 6 across the line y = x, so it is indeed the inverse. If your inverse doesn't produce a reflection across y = x, check your algebra and arithmetic for errors.