# Codomains of composite functions

• airborne18
In summary, when dealing with composite functions, the key aspect is that the codomain of the inner function and the domain of the outer function must be aligned. If not, the function will not be well-defined. Also, for function composition, the domain must be restricted to only the integers. It is important to note that the codomain is not necessarily equal to the range.
airborne18

## Homework Statement

Hopefully simple. Do composite functions have to have the same Codomain? What if they do not, does the smaller Codomain get canceled out? f(x) : R ->R g(x) :Z->Z f(x) g(x) : R->R Is this correct? Or do I need to hit the books a bit more?

## The Attempt at a Solution

The codomain of f(x) and g(x) are basically unrelated. If you want to define f(g(x)), the key aspect is that the codomain of g(x) and the domain of f(x) line up. If not, there will be values of x for which g(x) can't be plugged into f(x) and you fail to have a function. In such a case you have to restrict the domain of g(x) so that the codomain of g(x) is a subset of the domain of f(x)

Also, assuming that f(x) g(x) in your post means f(g(x)) (so we have function composition), then the domain has to be only the integers: you can't plug an arbitrary real number into g(x). Actually this is true even if you meant multiply the two functions

Office_Shredder said:
The codomain of f(x) and g(x) are basically unrelated. If you want to define f(g(x)), the key aspect is that the codomain of g(x) and the domain of f(x) line up. If not, there will be values of x for which g(x) can't be plugged into f(x) and you fail to have a function. In such a case you have to restrict the domain of g(x) so that the codomain of g(x) is a subset of the domain of f(x)

Also, assuming that f(x) g(x) in your post means f(g(x)) (so we have function composition), then the domain has to be only the integers: you can't plug an arbitrary real number into g(x). Actually this is true even if you meant multiply the two functions

Thanks. My assumption was that if there were values that were not valid for g(x), the function would simply be undefined for those values in R that are inconsistent with Z. That is what is screwing me up with this. My problem is that I always put Codomain = Range, and that has been a flaw in my thinking as well.

## 1. What is a codomain?

A codomain is the set of all possible outputs of a function. It is often denoted as the set of values that the function can map to.

## 2. How is a codomain different from a range?

A range is the set of actual outputs of a function, while a codomain is the set of all possible outputs. In other words, the range is a subset of the codomain.

## 3. Can a function have multiple codomains?

No, a function can only have one codomain. However, a codomain can contain multiple elements or values.

## 4. How do you determine the codomain of a composite function?

To determine the codomain of a composite function, you first need to find the range of the inner function. Then, you can use the range as the codomain for the outer function.

## 5. Why is understanding the codomain important in mathematics?

Understanding the codomain is important because it helps us to understand the limitations and possibilities of a function. It also allows us to define the domain and range of a function, which are essential concepts in mathematics.

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