Content

• Steps
• Tips

Math functions, commonly referred to as f (x) or g (x), can be thought of as the order in which mathematical operations are performed that go from "x" to "y". The inverse function f (x) is written as f (x). In the case of simple functions, it is not difficult to find the inverse function.

Steps

1. 1 Rewrite the function completely, replacing f (x) with y. In this case, "y" must be on one side of the function, and "x" - on the other. If you are given a function like 2 + y = 3x, you need to isolate "y" on one side and "x" on the other.
   • Example. We rewrite this function f (x) = 5x - 2 as y = 5x - 2... f (x) and "y" are interchangeable.
   • f (x) is the standard notation for a function, but if you are dealing with multiple functions, each of them will need to be assigned a different letter to make them easier to distinguish from each other. For example, functions are often referred to as g (x) and h (x).
2. 2 Find "x". In other words, do the math required to isolate the "x" to one side of the equal sign. Basic algebraic principles: if "x" has a numerical coefficient, then divide both sides of the function by this coefficient; if some free term is added to the term with "x", subtract it from both sides of the function (and so on).
   • Remember that you can apply any operation to one side of the equation only if you apply the same operation to all the terms on either side of the equal sign.
   • In our example, add 2 to both sides of the equation. You get y + 2 = 5x. Then divide both sides of the equation by 5 to get (y + 2) / 5 = x. Finally, rewrite the equation with "x" on the left: x = (y + 2) / 5.
3. 3 Change the variables by replacing "x" with "y" and vice versa. The result will be a function that is the opposite of the original one. In other words, if we plug in the x value into the original equation and find the y value, then by plugging that y value into the inverse function, we get the x value.
   • In our example, we get y = (x + 2) / 5.
4. 4 Replace "y" with f (x). Inverse functions are usually written as f (x) = (terms with "x"). It should be noted that in this case -1 is not an exponent; it is just notation for the inverse function.
   • Since "x" in the -1 power is equal to 1 / x, then f (x) is the notation 1 / f (x), which also denotes the inverse function of f (x).
5. 5 Check the work by substituting a constant value in the original function instead of "x". If you correctly found the inverse function by substituting the value "y" into it, you will find the substituted value "x".
   • For example, plug in x = 4. You get f (x) = 5 (4) - 2 or f (x) = 18.
   • Now plug 18 into the inverse and you get y = (18 + 2) / 5 = 20/5 = 4. That is, y = 4. This is the "x" plugged in, so you got the inverse correctly.

Tips

• When you perform algebraic operations on functions, you can freely substitute f (x) = y and f ^ (- 1) (x) = y in both directions. But writing the reverse function directly can be confusing, so stick with f (x) or f ^ (- 1) (x) to help you distinguish them from each other.
• Note that the inverse function is usually (but not always) a functional dependency.