Hello students, now let’s do few more problems on inverse of a function. The first one is this, we need to find inverse of this piece wise defined function where f is from 0, 4 to 0, 6. Okay, now let me give you the graph of this function so that you get a better idea. The graph looks like this. From 0 to 2, it’s x square, from 2 to 4 it’s x plus 2. Now, how do we backtrack. For 0 to 4 backtracking will be done by red curve and from 4 to 6 back tracking will be done by blue line. Now, say from 0 to 4 we will take 3 there. 3 will backtrack by red curve. While going it was square, so while coming back it should be square root. So, f inverse x here should be root of x. From 4 to 6 let us take 5, while going it was x plus 2, so, while coming back it should be minus 2, so x minus 2. Combining it we get our answer from 0 to 4 f inverse x is root x and from 4 to 6, f inverse x is x plus 2.

Moving ahead we have this example. We are given that fx is equal to its inverse and fx is kx plus 3. We got to find k. Simple, we will first find inverse and then equate it to fx to find k.

Now, for finding inverse, we will express x in terms of y as y is kx plus 3, x will be y minus 3 upon k and hence we have inverse equal to x by k minus 3 by k. Now as given f inverse x is equal to fx, so this should be equal to this. Now, comparing coefficients we have 1 by k equal to k and 3 equal to minus 3 by k. So, we get k equal to plus minus 1, from first equality and k equal to minus 1 from second equality. As there is ‘and’ in between them so taken into section we get k equal to minus 1. So, basically you did nothing you found f inverse x and equate it to fx to get k.

Now, let’s have another example. We got to find inverse of f where fx is  x plus 1 by x, okay. I will take it equal to y and try to express x in terms of y. It becomes quadratic and we get 2 values of x, one is this and the other is this. Now, what to do? We know inverse is unique if f is this one way, f inverse is also one way. So, f inverse should be unique. Which of the following is correct? Now, we will use if f is from a to b, f inverse will be b to a. So, f inverse should be from minus infinity minus 2 to minus infinity minus 1. So, taking y equal to minus 3, we find x. We get x equal to minus 0.3815 in this case. Approximately minus 0.38 and minus 2.61 in this case. Now, this belongs to minus infinity minus 1 whereas this does not belong. So, that expression should be wrong and this expression should be right. So, this is wrong and this is my inverse. So, f inverse x is x minus root of x square minus 4 equal to 2. Basically in this case we learnt two things, number one f inverse is unique, number 2 if we need to check which one is correct, use that f inverse will from b to a, okay.

Now, moving ahead we have this example. We got to find m for which this function becomes invertible. Now, don’t get confused, simple condition keep it in mind, very important. A function is invertible if and only if it is bijective. So, we need to know for which values of m this is bijective. Now, cubic polynomial always has range real, I told you. So, it is onto, so you have to only make sure that it is one-one. For one-one we know f dash should be either only positive greater than or equal to zero or only less than or equal to zero. It’s a quadratic with a positive. We can make this quadratic always greater than or equal to zero doing this. We know a condition for quadratic to be always positive. So, a is already positive taking D less than or equal to zero. We get m square is less than equal to 9. That is m lies between minus 3 to 3. So for these values of m it will be one-one. It is already onto be in a cubic polynomial. So, it is bijective and hence it is invertible. This is a good question spend some time as I used to say, spend some time on this question.

So, keep practicing, god bless, take care.