# Solving Functional Equation Homework

• zenos
In summary, the conversation discusses finding a solution for a functional equation involving a function f and its properties. The solution provided a rough sketch for the case where f is continuous and additional assumptions on f were discussed. The conversation also mentions the use of Cauchy's exponential equation and references a previous written work on functional equations.
zenos

## Homework Statement

Is the solution correct

## The Attempt at a Solution

all are in the file

#### Attachments

• file.pdf
17.9 KB · Views: 207
Last edited:
That does not look like a complete solution. For one, f(x) = 1 for all x and f identically equal to 0 are trivial solutions, and these can be cited by inspection. I'll keep trying things, but it would help if there were any additional assumptions on f, such as continuity perhaps?

All right, here is a rough sketch for the case where f is continuous. As before, f identically equal to zero is a trivial solution. Now suppose there exists a real number c for which f(c) =/= 0. Then

$$f(x)f(c) = f(\sqrt{x^2 + c^2}) = f(-x)f(c).$$

This implies that f(x) = f(|x|) for all real x. Define $g(x) = f(\sqrt{x})$ for $x \geq 0$. Note that g satisfies Cauchy's exponential equation: g(x + y) = g(x)g(y) for $x,y \geq 0.$

Now see if you can complete the argument based off of the proof for Cauchy's exponential equation. For reference, attached is something I wrote awhile ago when I was still interested in functional equations.

#### Attachments

• cauchyexp.doc
31.5 KB · Views: 182

## What is a functional equation?

A functional equation is an equation in which the unknown quantity is a function. It involves a set of variables and a rule for combining them to produce a result. In other words, it is an equation that describes the relationship between a function and its input values.

## Why are functional equations important?

Functional equations are important because they allow us to model and solve real-world problems using mathematical concepts. They are also used in various fields of science, including physics, economics, and engineering, to describe and analyze different phenomena.

## What are the steps to solving a functional equation homework?

The steps to solving a functional equation homework are as follows:

1. Identify the variables and their relationships in the equation.
2. Apply any given conditions or constraints to the equation.
3. Use algebraic manipulation or substitution to simplify the equation.
4. Find the solution(s) to the equation.
5. Check for extraneous solutions and ensure that the solution(s) satisfy the given conditions.

## What are some common types of functional equations?

Some common types of functional equations include linear, quadratic, exponential, logarithmic, and trigonometric equations. Other types include functional equations involving derivatives or integrals, and functional equations with multiple variables.

## What are some tips for solving functional equation homework?

Here are some tips for solving functional equation homework:

• Read the problem carefully and understand what is being asked.
• Make sure you are familiar with the basic properties and rules of the functions involved.
• Use graphs or tables to visualize the relationship between the variables.
• Try different approaches and methods to solve the equation.
• Check your solution(s) and rework the problem if necessary.

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