To prove that the composition of mappings is surjective, one must show that for every element in the codomain of the final mapping, there exists at least one element in the domain of the initial mapping that maps to it. This can be done by composing the mappings and demonstrating that the composition covers the entire codomain. To prove that the composition is injective, one must show that distinct elements in the domain of the initial mapping map to distinct elements in the codomain of the final mapping. This can be achieved by assuming two elements in the domain that map to the same element in the codomain and then showing that they must be the same element. **

To prove the composition of mappings as surjective, we need to show that for every element in the codomain of the final mapping, there exists an element in the domain of the initial mapping that maps to it. In other words, we need to show that the composition mapping covers the entire codomain. To prove the composition of mappings as injective, we need to show that if two elements in the domain of the initial mapping map to the same element in the codomain, then the elements themselves are the same. This means that the composition mapping does not "lose" any information from the initial mapping. **

To determine if a function is injective, we need to check if each input value maps to a unique output value. If the function f(x) = x^2 is defined on the set of real numbers, then it is not injective because multiple input values (e.g. 2 and -2) map to the same output value (4). Therefore, the function f(x) = x^2 is not injective. **

One way to prove that function f is injective if function g is injective is to show that for any two distinct inputs x1 and x2, the outputs f(x1) and f(x2) are also distinct. Since g is injective, we know that g(x1) and g(x2) are distinct, and we can use this property to show that f is injective as well. Specifically, we can use the fact that g(f(x1)) = g(f(x2)) implies f(x1) = f(x2), and since g is injective, this implies x1 = x2. Therefore, f is injective. **

To show that if f and g are injective, then gf is also injective, we can use the definition of injective functions. An injective function is one where distinct inputs map to distinct outputs. So, if f and g are injective, then for any distinct inputs x1 and x2, f(x1) ≠ f(x2) and g(y1) ≠ g(y2) for any distinct outputs y1 and y2. Now, consider the composition gf. If gf(x1) = gf(x2), then f(x1) = f(x2), which implies x1 = x2 by the injectivity of f. Therefore, gf is also injective. **

To determine if a mapping is surjective or injective, we need to look at the properties of the mapping. Please provide the specific mapping you would like me to analyze. **

The first mapping is injective because each element in the domain is mapped to a unique element in the codomain. The second mapping is surjective because every element in the codomain is mapped to by at least one element in the domain. **

To determine if a function is injective or surjective, we need to look at its properties. A function is injective if each element in the domain maps to a unique element in the codomain, meaning no two different elements in the domain map to the same element in the codomain. A function is surjective if every element in the codomain is mapped to by at least one element in the domain. To determine if a function is injective or surjective, we can analyze its graph, its algebraic representation, or its properties. If the function passes the horizontal line test, it is injective. If every element in the codomain has at least one pre-image in the domain, the function is surjective. If the function is both injective and surjective, it is bijective. **