Proving the Identity Function: Composed Functions

Thread starter andmcg
Start date Jan 30, 2011
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Function Functions Identity

In summary, the identity function is a mathematical concept where the output of a function is equal to its input. To prove this function, one must show that the output is equal to the input for all possible values. A composed function is made up of two or more functions and to prove that two composed functions are equal, one must show that their outputs are equal for all possible inputs. Proving the identity function for composed functions is significant because it helps to understand the relationship between different functions and their properties.

Jan 30, 2011

andmcg

Suppose that f composed with g equals g composed with f for all functions g. Show that f is the identity function.

I really just don't know where to start.

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Jan 30, 2011

Fredrik

Staff Emeritus

Science Advisor

Gold Member

*Deleted*

That was probably a bad tip. I'll try to think of something better.

OK, here we go...

If f isn't the identity, there's an x such that f(x)≠x. Now choose a g that contradicts that.

Last edited: Jan 30, 2011

FAQ: Proving the Identity Function: Composed Functions

What is the identity function?

The identity function is a mathematical concept where the output of a function is equal to its input. It is represented by the symbol "id" or "I" and can be thought of as a "do nothing" function.

How do you prove the identity function?

To prove that a function is the identity function, you must show that the output of the function is equal to its input for all possible values. This can be done by substituting a variable for the input and using algebraic manipulations to show that the output is the same as the input.

What is a composed function?

A composed function is a function that is made up of two or more functions. This means that the output of one function is used as the input for another function, creating a chain of functions.

How do you prove that two composed functions are equal?

To prove that two composed functions are equal, you must show that their outputs are equal for all possible inputs. This can be done by substituting a variable for the input and using algebraic manipulations to show that the outputs are the same for both functions.

What is the significance of proving the identity function for composed functions?

Proving the identity function for composed functions is important because it allows us to understand the relationship between different functions and how they interact with each other. It also helps us to better understand the properties and behavior of these functions, which can be useful in solving more complex mathematical problems.