f((x^2 - 1)/x) = x^2 - x | Which is f(x)

[00:00:02]Good day, viewers. You are welcome.

[00:00:06]We are given f of x squared minus one over two equals x squared minus x.

[00:00:15]We are interested to find f of x. We are looking at this problem.

[00:00:21]This is not a single-valued functions.

[00:00:25]I mean, it's not one-to-one.

[00:00:28]We are getting two answers.

[00:00:31]One will be positive, one will be negative. But for the case of checking, which of these option are we going with?

[00:00:36]You can pause this video and give it a try.

[00:00:40]So, let's solve this step by step.

[00:00:48]So, from this domain, we have x squared minus one divided by two.

[00:00:56]Of course, I let this one to be equals u.

[00:00:59]But I have to replace this with another variable.

[00:01:04]So, the equation becomes f of u equals x squared minus x.

[00:01:13]But we have to express the whole of this in terms of u.

[00:01:17]So, let's come back here.

[00:01:20]So, if I cross multiply, we have x squared minus one equals 2u.

[00:01:27]And by making x squared the subject of the formula, we are going to have x squared equals 1 plus >> [snorts] >> 2u.

[00:01:39]So, I'm going to start this. And coming back to where we have f of u equals x squared minus x.

[00:01:50]>> [snorts] >> So, here we have x squared. If I replace it with 1 + 2u.

[00:01:57]>> [snorts] >> We have f of u equals 1 + 2u x.

[00:02:07]Well, this x must not be here because everything must be in terms of u.

[00:02:12]Let's try to look at how we can get x.

[00:02:16]Again, we come back to x squared equals 1 + 2u.

[00:02:21]From then we are going to take the square root of both sides for 1 + 2u.

[00:02:30]So, taking the square root of this, you know, we have plus or minus x.

[00:02:35]Our square and square root cancel each other. We have only x equals plus or minus square root of 1 + 2u.

[00:02:44]But I said earlier that this is not a single valued function.

[00:02:49]We have plus or negative x, but only one of it satisfy and we are going to show how only one of it satisfy by putting a quick check of um the answer we get.

[00:03:04]So, let's replace this in f of u equals 1 + 2u So, the x now is plus or minus square root of 1 + 2u.

[00:03:19]So, we are going to put this in bracket and this is plus or minus square root of 1 + 2u.

[00:03:26]By expanding this we have 1 + 2u plus times minus times plus, that is minus.

[00:03:36]And here we have minus times minus, this gives us plus.

[00:03:39]We have the square root of 1 + 2u.

[00:03:44]So, there are two solutions.

[00:03:47]So, the first one is f of u equals 1 + 2 u plus square root of 1 + 2 u.

[00:03:59]And again, we have f of u equals 1 + 2 u minus square root of 1 + 2 u.

[00:04:08]So, we're going to look at which of these really satisfy the uh is is truly the answers.

[00:04:17]Is it the one with positive or the one with negative?

[00:04:22]So, before proceeding, let's try to change this one to x because u and x are both members of real number.

[00:04:28]We have f of x equals 1 + 2 x + square root of 1 + 2 x.

[00:04:36]Why is this not a single-valued function? Suppose I assume that x equals to zero.

[00:04:43]We're going to have two different values.

[00:04:45]We have this as f of zero equals 1 + 2 * 0. This gives zero plus square root of 1 + 0.

[00:04:56]So, this is 1 + 1. This gives us two.

[00:05:00]When we check for the positive, and again, we check for the negative, we have f of x equals 1 + 2 x minus square root of 1 + 2 x.

[00:05:16]So, with the same value of x equals to zero, we get different answers.

[00:05:20]So, let's suppose this is f of zero equals 1 + 0 minus square root of 1.

[00:05:25]And this gives us 1 minus 1, and 1 minus 1 equals zero.

[00:05:30]So, there are two different outputs.

[00:05:33]Two and a zero. This is not a single-valued function. But, which of these is the solution to this problem?

[00:05:40]Let's try to verify.

[00:05:44]So, let's put uh a quick check on the first one.

[00:05:48]Since we have f of x squared minus one over two, uh, initially it was f of x.

[00:05:55]So, anywhere we see x, we replace it with x squared minus one over two.

[00:06:01]So, with this we have one plus two into bracket of x squared minus one over two, then plus the square root of one plus two into bracket of x squared minus one over two.

[00:06:17]So, this equals two cancel two, and we have one plus x squared minus one plus the square root of so, two also cancel two here.

[00:06:28]And this gives us one plus x squared minus one.

[00:06:32]So, plus one minus one cancel, plus one minus one also cancel here.

[00:06:36]So, this gives us x squared plus the square root of x squared.

[00:06:42]So, for square and square root cancel each other, we then have f of x squared minus one over two equals x squared plus x. And this does not give us the answer because the question says x squared minus x.

[00:07:00]And likewise, if you check for the second one, where we have f of x equals one plus two x minus square root of one plus two x.

[00:07:14]So, replacing x with x squared minus one over two, we have this as one plus two into bracket of x squared minus one over two minus square root of one plus two into bracket of x squared minus one over two.

[00:07:33]So, [snorts] for two cancel two, and we have one plus x squared minus one minus square root of one plus two also cancel two here.

[00:07:44]We have x squared minus 1.

[00:07:46]So for one cancelled one and also the one in the bracket cancelled each other.

[00:07:50]So we have x squared minus square root of x squared that is x. So which shows that f of x squared minus 1 over 2 for the negative one is the only solution. So we end by go for the second one with negative not the first one.

[00:08:10]So this how to solve this problem.

[00:08:11]Thanks for watching. See you in the next video. Never stop learning. Bye-bye.