If both functions f and g are surjective, what can be concluded about the composition g ◦ f?
Understand the Problem
The question is asking about the properties of the composition of two functions, specifically whether the composition of surjective functions is also surjective. Understanding the definitions of surjective and the rules of function composition will help determine the correct conclusion.
Answer
The composition g ◦ f is surjective.
If both functions f and g are surjective, the composition g ◦ f is also surjective.
Answer for screen readers
If both functions f and g are surjective, the composition g ◦ f is also surjective.
More Information
When both f and g are surjective, every element in the codomain of g has a pre-image in the domain of f. Thus, the composition g ◦ f is surjective because for every element in the codomain of g, there's an element in the domain of g (and, consequently, in the domain of f) that maps to it.
Tips
A common mistake is assuming that the composition is bijective or injective, but surjectivity alone does not imply these properties.
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