One to one and onto in composite function

In summary, the fact that f(g(x)) is onto does not guarantee that both f(x) and g(x) are onto, and the fact that it is one to one only guarantees that f(x) is one to one. This can be seen in the example of f(x) = (x,0) and g(x,y) = x+y.
  • #1
BlackDeath
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Homework Statement



I just want to make sure that I am correct. if we have a composite function f(g(x)).

Homework Equations


f(g(x)) is onto if and only if both f(x) and g(x) are onto
f(g(x)) is one to one if and only if or both f(x) and g(x) are one to one


The Attempt at a Solution



when I try to make f(x) is onto, but not one to one. And g(x) is one to one but not onto, f(g(x)) is not onto
 
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  • #2
This is false: The fact that f(g(x)) is one to one only guarantees that f(x) is one to one.

For example, let [itex] f: \mathbb R \to \mathbb R^2 [/itex] by [itex] f(x) = (x,0) [/itex] and [itex] g: \mathbb R^2 \to \mathbb R [/itex] be [itex] g(x,y) = x+y [/itex]. f is one-to-one but g is not one-to-one. However, the function g(f(x)) = x is just the identity function and is injective.

Precisely the same example shows that this does not work for onto functions. g(x,y) is onto but f(x) is not. However, g(f(x)) = x is onto, so again it breaks.
 

Related to One to one and onto in composite function

What is a one-to-one function?

A one-to-one function is a type of function in which each element of the domain maps to a unique element in the range. This means that for every input, there is only one possible output. In other words, no two elements in the domain will have the same output.

What is an onto function?

An onto function is a type of function in which every element in the range is mapped to by at least one element in the domain. This means that for every output, there is at least one input that maps to it. In other words, the range of the function is equal to its codomain.

What is a composite function?

A composite function is a function that is formed by combining two or more functions. The output of one function becomes the input of the next function. In other words, the output of the first function is substituted into the second function, and the resulting output is the final output of the composite function.

How do you determine if a composite function is one-to-one?

If a composite function is formed by combining two one-to-one functions, then the resulting composite function will also be one-to-one. However, if one of the functions is not one-to-one, then the resulting composite function may not be one-to-one.

How do you determine if a composite function is onto?

If a composite function is formed by combining two onto functions, then the resulting composite function will also be onto. However, if one of the functions is not onto, then the resulting composite function may not be onto. Additionally, if the range and codomain of the composite function are not equal, then the composite function is not onto.

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