Inverse of a Function

Inverse of a Function

In this Post I will discuss the method for finding the inverse of a given function from the 4 options as given in the Entrance tests of Engineering Universities.

The very first step before finding the inverse is to make a check whether the function is One-to-One or not. 

By One-to-One function we mean that function for example f(x) should not have the same value at two different values of x. In other words for each element of the Domain there should be a distinct element in the range.

Check for One-to-One:

A function which passes the horizontal line test is a One-to-One function. This means if a horizontal line is drawn it should not cross the function curve at more than one point.

Another way to check if a function is One-to-One or not is shown below:

If ,

f(a) = f(b) , implies that a=b , the given function is One-to-One.

Consider the following examples:

f(x) = 5x + 10

f(a) = 5a + 10

f(b) = 5b + 10

Taking f(a) = f(b) gives,

5a + 10 = 5a - 10

this means

a=b

Thus the given function is One-to-One.

Now consider f(x) = x^2 + 1

f(a) = a^2 + 1

f(b) = b^2 + 1

Taking f(a) = f(b) gives,

a^2 + 1= b^2 + 1

this means

a^2 = b^2 

Now here are two possibilities ,

i.e

a=b or a=-b

Hence the given function is not One-to-One. 

I hope its clear. 

So first step before finding the inverse is to make the check for One-to-One function. If the given function is not One-to-One its inverse won't exist. In such cases there will be an option like "does not exist", "none of the above" etc. So it will be the correct answer.

In case the function is One-to-One follow the steps below to find the inverse.

The composition of a function and its inverse function is always equal to x. So take the composition of the given function with the options provided to you. The option which will give you the x will be the answer. Remember that answer of composition must be x, some people confuse it with 1 in the exams. 

This process is useful when the function to be dealt with is somewhat complex. In case of simple functions you can follow the traditional method as per mentioned in your text books. In case you forget the method to find the inverse, this process will come in handy. I'll suggest you to use this method in exams as it decrease the chances of mistakes.

In case of any confusion feel free to ask.