An Oxford University Test Question | Can you solve ?

追加: 2026-05-05

[00:00:01]Hello, welcome back once again. Today we have this interesting functional problem.

[00:00:06]We're given that f of f of x is equal to x plus one. In this video, we're going to find this functional equation with this property. So, let's get started.

[00:00:17]Now, take a look at the right-hand side here. The highest degree of x here is one. So, this must be a linear function.

[00:00:26]But, let us consider the following.

[00:00:28]Say x I mean f is equal to let's say we have x.

[00:00:34]And this is a linear function, right?

[00:00:36]So, let us compute f of x.

[00:00:40]So, here you would see that here f of f of x is going to be equal to this x, replace it with x. So, that will also give us x.

[00:00:52]Okay? So, let's say f is equal to 2x plus one. This one here is a linear function. So, let us compute f of f of x. So, here f of f of x is equal to two into bracket. So, this x replace with 2x plus one. So, get 2x plus one and then plus one. So, if you simplify this carefully, we see that f of f of x here is equal to 4x plus three.

[00:01:26]And you can see the degree of x here is one. So, therefore, this functional equation must be a linear function. So, let's get started.

[00:01:37]Now, we're given that f of f of x is equal to x plus one.

[00:01:46]So, let's f of x equals ax plus b since f is a linear function.

[00:01:58]So, therefore, here f of f of x is going to be equal to a into bracket.

[00:02:07]So, replace x with ax plus b.

[00:02:10]So, that will be ax plus b.

[00:02:14]And then plus b.

[00:02:17]So, here we see f of f of x is equal to here it will be a squared x plus ab.

[00:02:30]Then plus b.

[00:02:34]Now, let us recall again that this left-hand side is equal to x plus one.

[00:02:40]So, therefore, we have a squared x plus ab plus b is equal to x plus one. Now, let's compare coefficients.

[00:02:53]So, here we see that a squared is equal to the equation of x here, which is one, right? So, this will give us a is equal to one or a is equal to negative one.

[00:03:06]So, let's compare the constants, right?

[00:03:09]So, here So, we can factor out b. So, you get b into bracket a plus one is equal to one. Now, you can see that if a is equal to one, we get 2b is equal to one, which is given us b is equal to one over two. But, when a is equal to negative one, we get zero. B times zero is zero, so zero is equal to one is a false statement. So, there's no solution with a is equal to negative one. So, therefore, the functional equation with the above property is equal to So, that is x plus one over two. And this right over here is our correct answer. Thank you for watching. If you enjoyed the video, please kindly subscribe to this channel.

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