How to Find the Inverse of a Function

Inverse of a function can be understood as the value of Y for which X is a function. We refer a general mathematical function as f(x). Similarly, in an equation it will be represented as y = f(x). For any value of X, you will get an equivalent value of Y. Now, if we write the function as f^-1(x), it is represented as the inverse of the function. For any value of Y, we will then get a similar value of X back. Hence, the inverse of the function can be defined as the returning value of X when the value is put back on Y set. Here is how you can find the inverse of a function easily.

Writing the equation for Inverse function

To solve and get the inverse of a function easily, we would be required to write the function first. To make it easy, we replace the f(x) with Y and then solve for the value of x in order to get the value of inverse. On one side of the formula would be Y and the other side would contain all other values with X. in other case, if the equation is already written in X and Y, then we need to isolate Y on one side and bring all other elements on the other side. For example, if the equation is written like 3 + y = 3x + 9, then we need to rewrite the equation as y = 3x + 6. This would the equation after isolating Y.

Again if the equation is written as f(x) = 3x + 6, then replace f(x) it with Y. The equation would now look like, Y = 3x + 6. F(x) is the standard notation for a function. If there are other functions, we would write them as g(x) and h(x). So, solving them becomes hard. Hence we change the function notation to Y and then solve for Y.

Then again, solving for X in that equation would important. After performing the necessary mathematical operations on the function, isolate X by putting it one side of the equation. If X has a numeric coefficient, you should divide the other side with that coefficient. If a  number is added to X, then subtract the number from other side or vice versa to isolate the equation. Lets say we have an equation in the form of y = 3x + 6, so when you isolate the equation for X, you get the equation as x = (y – 6)/3.

Now switch the variables in the equation. This means that you require replacing the X with Y and subsequently Y with X in the equation. The resulting equation would be the inverse of the function you were seeking for. In other words, if we replace the value of Y in the original equation, we would get a value for X which would be the inverse of the function.

So, we now have equation as x = (y-6)/3, this equation would become, y = (x-6)/3. The value of Y here would be the inverse of the function. Now, for the last part of it, you will be required to replace the value of Y with “f^-1(x).” the inverse of a function is generally written as “f^-1(x) = terms in X. you should not worry about the exponent here as it is just a representation of the function. It is to show that the following text in X is the inverse of the function f(x).  “f^-1(x) can also be thought as the 1/f(x) of the function which is then called as the Inverse of the function.

So, now the equation we had was y = (x-6)/3. Replacing the Y with f^-1(x) would make the equation as f^-1(x) = (x-6)/3. Hence this is the Inverse of the function. So, now is the time to evaluate the correctness of the inverse you just found out. To find out if the inverse is correct, try substituting a constant into the original function of x. if what you found out was correct, then plugging the result into the inverse function will yield you the correct result for inverse. For example, lets’ place a value such as such as 9 into the equation; this would give us the inverse value as 1 for the equation. This is how we find the inverse of any function.