If f(x+y)=f(x)+f(y) and f(1)=3, find f(2), f(3) and f(4) . Is f(x+y)=f(x)+f(y) for all functions

If f(x+y)=f(x)+f(y) and f(1)=3, find f(2), f(3) and f(4) ....

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Key Concepts

Linearity

Linearity is the property of a function where it can be written in the form f(x) = cx for some constant c. In the context of the equation f(x+y) = f(x) + f(y) and a given value such as f(1), one can deduce that the function is linear if regularity conditions are assumed.

Functional Equations

A functional equation involves determining a function that satisfies certain conditions imposed on its values. In the context of f(x+y)=f(x)+f(y), solving the functional equation entails investigating the nature of the function and determining whether additional conditions are necessary to ensure a unique and well-behaved solution.

Cauchy's Functional Equation

Cauchy's functional equation, given by f(x+y) = f(x) + f(y), is a classic example in the study of functional equations. It generally characterizes additive functions, and under additional conditions like continuity or boundedness, it implies that the function is linear.

Additive Functions

Additive functions are functions that satisfy the property f(x+y) = f(x)+f(y) for all inputs. When these functions are also subject to regularity conditions, they can be expressed as linear functions; however, without such conditions, there may exist non-linear, pathological solutions.

Transcript

00:01 All right, let's explore what is somewhat of a postulate or a truism.

00:06 And we want to see when is it true, when is it not true? and i'm referring to f of x plus y is equal to f of x plus f of y.

00:18 And we're going to explore this with an example.

00:23 So if we know that f of 1 is equal to 2, then we want to find, f of 2, f of 3, and f of 4.

00:41 So how can we tell what those are based on our knowledge of f of 1? so keep in mind, f is any function.

00:49 It's a function, right? and it's a function that has an output of 2.

00:54 So we could draw this, right, an f of x.

01:00 And so far we know that in this table, when we input of 1, we're going to get out a 2.

01:05 So why not make f of x equal to 2x? that means that 1 times 2 would be 2.

01:17 And this y here can be any number.

01:27 So although that might seem a little confusing, like y should be the second column, it's not.

01:33 In this situation, we're talking about the function of one number plus another number is equal to the function of the first number plus the function of the second number when we're referring to the same function.

01:46 Now, f of 1 equal to 2 means that f of x equal to 2x could be it.

01:53 All right.

01:54 If that's the case, then that means that f of 2 is going to be 4, f of 3 is going to be 6, and f of 4 is going to be 8...

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