Injectivity is a property of a function that means each input has a unique output. In other words, no two different inputs can have the same output.
Surjectivity is a property of a function that means every element in the range of the function has at least one corresponding element in the domain. In other words, every possible output value is mapped to by at least one input value.
To prove injectivity of a composite function f, you need to show that for any two distinct inputs x1 and x2, the outputs f(x1) and f(x2) are also distinct. This can be done by assuming that f(x1) = f(x2) and then using the definition of a composite function to show that x1 = x2, thus proving that the function is injective.
To prove surjectivity of a composite function f, you need to show that for every output value y, there exists at least one input value x that maps to y. This can be done by choosing an arbitrary y and then using the definition of a composite function to find an x that maps to y, thus proving that the function is surjective.
The main difference between injectivity and surjectivity is that injectivity focuses on unique outputs for each input, while surjectivity focuses on every possible output being mapped to by at least one input. In other words, injectivity ensures that no two different inputs have the same output, while surjectivity ensures that no outputs are left unmapped.