A Singaporean Function Problem | Can you solve ?

Añadido: 2026-05-05

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

[00:00:07]We're going to find f of x given f of 4x f of x is equal to 9x.

[00:00:17]Now, this functional equation looks like a linear function, right? Here, the highest degree of x here is 1.

[00:00:27]So, we assume that this function is a linear function, okay? So, therefore, we're going to let f of x equals ax + b.

[00:00:42]So, here a is a real number greater than zero, and here b is a constant. So, b is a constant.

[00:00:53]So, here, since we have f of x in terms of this a and b as ax + b, therefore, f of 4x, according to the problem, so f of 4x is going to be equal to replace x with 4x, so we get a into bracket 4x, then + b.

[00:01:12]So, this is going to be equal to 4ax + b, awesome.

[00:01:18]Then, f of x from the equation, remember our f of x is already this. So, therefore, the following property of this functional equation, that is f of 4x - f of x, which is equal to 9x, in terms of a and b, is going to be 4 ax + b - here ax + b.

[00:01:42]This is equal to 9x.

[00:01:45]So, from here, 4ax + b ax, then - b is equal to 9x.

[00:01:56]From here, let's simplify. So, 4ax - ax, then + b - b is equal to 9x.

[00:02:06]So, from here, 4ax - ax is 3ax.

[00:02:10]Then, b - b is 0. This is equal to 9x.

[00:02:14]Now, here, we're going to equate coefficients, right? So, it implies from here that 3a is equal to 9. To solve for a, divide both side by the coefficient of a, which is 3. 3 and 3 gets cancelled. And from here, we have a is equal to 3.

[00:02:35]So, therefore, our functional equation now, f of x is equal to, remember, it is ax + b. a is equal to 3, that becomes 3 x + b. Here, b is a constant, and this right over here is our solution. So, f of x is equal to 3x + b. So, when you check out f of 4x - f of x here, it is going to be equal to 9, and that's is the property of this functional equation. So, hence here, we have our solution. B is a constant, b is a real number. Thank you for watching. If you enjoyed this video, please kindly subscribe to this channel. Also, like, comment, and share.