A mathematical function (usually expressed as f (x)) can be interpreted as a formula that allows you to derive the value of y based on a given value of x. The inverse function of f (x) (which is expressed as f-1(x)) is in practice the opposite procedure, thanks to which the value of x is obtained once that of y has been entered. Finding the inverse of a function may seem like a complicated process, but knowledge of basic algebraic operations is enough for simple equations. Read on to learn how to do it.
Steps
Step 1. Write the function by replacing f (x) with y, if necessary
The formula should appear with y, alone, on one side of the equality sign and the terms with x on the other side. If the equation is written with the terms of y and x (for example 2 + y = 3x2), then you have to solve for y by isolating it on one side of the "equal" sign.
- Example: consider the function f (x) = 5x - 2, which can be written as y = 5x - 2 simply replacing "f (x)" with y.
- Note: f (x) is a standard notation to indicate a function, but if you are dealing with multiple functions, each of them will have a different letter to make identification easier. For example, you can write g (x) and h (x) (which are equally common letters for writing a function).
Step 2. Solve the equation for x
In other words, perform the necessary mathematical operations to isolate x on one side of the equality sign. In this step, the simple algebraic principles will help you. If x has a numerical coefficient, divide both sides of the equation by that number; if x is added to a value, subtract the latter on both sides of the equation and so on.
- Remember to do the operations on both terms on either side of the equal sign.
- Example: we always consider the previous equation and add the value of 2 on both sides. This leads us to transcribe the formula as: y + 2 = 5x. Now we should divide both terms by 5 and we will get: (y + 2) / 5 = x. Finally, to make reading easier, we bring the "x" to the left side of the equation and rewrite the latter as: x = (y + 2) / 5.
Step 3. Replace the variables
Change x to y and vice versa. The resulting equation is the inverse of the original one. In other words, if you enter the value of x in the initial equation and get a certain solution, when you enter this data in the inverse equation (always for x) you will find the starting value again!
Example: after replacing x and y we get: y = (x + 2) / 5.
Step 4. Replace y with "f-1(x) ".
Inverse functions are usually expressed with the notation f-1(x) = (terms in x). Note that, in this case, the exponent -1 does not mean that you have to perform a power operation on the function. It is only a conventional spelling to indicate the inverse function of the original.
Since raising x to -1 leads you to a fractional solution (1 / x) then you might think that f-1(x) is a way of writing "1 / f (x)" which means the inverse of f (x).
Step 5. Check your work
Try replacing the unknown x with a constant in the original function. If you have done the steps correctly, you should be able to enter the result in the inverse function and find the starting constant.
- Example: we assign the value 4 to x within the starting equation. This brings you to: f (x) = 5 (4) - 2, so f (x) = 18.
- Now we replace x of the inverse function with the result we just found, 18. So we will have that y = (18 + 2) / 5, simplifying: y = 20/5 = 4. 4 is the original value we assigned to x, so our inverse function is correct.
Advice
- You can freely switch between f (x) = y and f ^ (- 1) (x) = y notation without any problems, when you are performing algebraic operations on your functions. However, it can be confusing to keep the original function and the inverse function in the direct form; it is better to use the notation f (x) or f ^ (- 1) (x), if you are not using either function, which helps to distinguish them better.
- Note that the inverse of a function is usually, but not always, also a function.
