f(4x) - f(2x) = x^2 | f(x) = ?

[00:00:01]Good day, viewers. You are welcome.

[00:00:04]Here we have an interesting functional equations given that f of 4x - f of 2x = x ^ 2. And let's see how we can find f of x.

[00:00:16]So, at this side we have a quadratic because of the power of two.

[00:00:21]And here we have 4x. So, here we have 2x. So, which is more of a polynomial functions.

[00:00:27]>> [snorts] >> To solve this, let's try and assumed a solution of polynomial for a quadratic specifically because this is a power of two.

[00:00:39]Then assume that if f of x equals ax ^ 2 plus bx plus c.

[00:00:50]Then, I need to find f of 4x and f of 2x.

[00:00:57]For this, we have f of 4x.

[00:01:00]This is equals anywhere I see x, I'm replacing it with 4x. Here becomes a into bracket of 4x raised to power 2 plus b into bracket of 4x plus c.

[00:01:15]Then, we have a into bracket of 4x ^ 2, that gives us 16x ^ 2 plus 4bx plus c.

[00:01:30]And which we also write this as 16ax ^ 2 plus 4bx plus c.

[00:01:42]Again, we need to find f of 2x. For f of 2x, we are going to have a into bracket of 2x ^ 2 plus b into bracket of 2x plus c.

[00:01:58]Then, the whole of this becomes 4 ax squared + 2bx + c.

[00:02:09]So, after getting this, we have to combine these together as f of 4x f of 2x = x squared.

[00:02:20]Where our f of 4x is given as 16 ax squared + 4bx + c.

[00:02:32]Then, minus open bracket, our f of 2x is given as 4ax squared + 2bx + c.

[00:02:44]And everything equals x squared.

[00:02:49]By combining these together, 16ax squared - 16ax squared, this gives us 12ax squared.

[00:02:58]Then, here is 4bx and a -4bx, which gives us 2bx.

[00:03:06]And here we have c - c, that cancels out.

[00:03:10]And everything equals x squared.

[00:03:15]Then, we have to do the comparisons.

[00:03:18]Here we have x squared, and this is also x squared.

[00:03:22]So, by combining by comparing this with this, and also we look for what is equals here. This is just like when we have 0x. So, 0 * x, that is 0. It means that we combine these, everything must be equals 0.

[00:03:39]And for this one, its x squared coefficient here is 1. That is, the 12a must be equals 1.

[00:03:47]And if 12a equals 1, then a equals 1/12.

[00:03:53]So, while for B X, we are going to have 2 B will be equal zero. So, this shows that our B here equals zero.

[00:04:04]Then, for f of x, we are going to substitute all these thing back into the assumed function above, which is f of x.

[00:04:16]By substituting in this, we have f of x equals a x squared plus b x plus c.

[00:04:27]But, a here is given as 1/12 and b here equals zero.

[00:04:33]By replacing this, this gives f of x equals 1/12 x squared plus zero multiplied by x plus c.

[00:04:48]Here we have f of x equals x squared over 12 plus c.

[00:04:56]So, this is what we have for f of x.

[00:05:00]But, what is the value of c here? C is any constant values.

[00:05:05]Any constant value is the value of is what we have here as c.

[00:05:09]But, to be more robust on these functional equations, let's consider another method to make it a general form.

[00:05:20]So, let's start iterating.

[00:05:22]>> [snorts] >> So, calling this one method method two, so which is uh more of related to the first method.

[00:05:32]We have f of 4 x minus f of 2 x. This equals x squared.

[00:05:44]Suppose I apply substitution method here.

[00:05:48]Let's say I replace X here with T divided by 2.

[00:05:53]Let's assume this.

[00:05:55]So, by replacing X with T over 2, we're going to have here as F of 4 multiplied by T over 2 F of 2 multiplied by T over 2 = T divided by 2 all raised to power of 2.

[00:06:14]So, to reduce this as we have F of 2T F of So, this cancels and we have F of T equals this is T raised to power 2 divided by 4.

[00:06:30]So, this is what we have at this point.

[00:06:33]So, here we have F of 2T, which is almost resemble the original equations because here is F of 2T and they follow the same patterns with the first one.

[00:06:45]Suppose I replace this uh F of T if I replace it with G of T, for instance, let's have another functions.

[00:06:54]So, here becomes G of 2T G of T = T squared divided by 4.

[00:07:06]Then, we can now assume the general solution of G of T which equals A T squared + B T + C.

[00:07:22]And when we solve this one as well we're going to get something of what we get initially. It's just like this is only going to change a little bit from what we have uh from the beginning.

[00:07:37]So, [snorts] this is more robust than the first one that we did.

[00:07:42]And that we get the same answer applying this one or applying the first one.

[00:07:48]And we can also try and verify if the left hand side and the right hand side and are equal by substituting the value of uh f of x that we got at this point in back into the given function and we are going to get the result. And also to verify that c here is always constant, so we just try and keep assuming that let's say our x here equals to zero.

[00:08:14]So, we try and substitute and check what we have at the two side. I mean substitute this into f of 4x, this becomes f of zero minus f of zero, everything equals zero.

[00:08:26]So, all these they are constant, so which shows that zero here equals zero.

[00:08:31]So, I still test for another one, let's say at x equals five, so we get another different things. So, this how to solve these functional equations. See you in the next video and don't forget to subscribe. Bye-bye.