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When a function is injective, it means that each element of the function's domain maps to a distinct element in the codomain. In simpler terms, different inputs will always lead to different outputs. Consider a scenario where you input values into a function, and regardless of which values you choose, the resulting outputs are always unique. That's the essence of injectiveness.

To check if a function is injective, you can employ the horizontal line test for functions depicted graphically. If a horizontal line cuts the curve of the function in more than one place, the function is not injective. Alternatively, if you solve the equation \(F(x_1) = F(x_2)\) as in our example, and it results in \(x_1 = x_2\), then the function is injective. This ensures that distinct inputs get mapped to distinct outputs.

A function is surjective if every element in its codomain has a pre-image in the domain. Essentially, a surjective function will cover the entire range of the codomain, leaving no gaps. This property is also known as being "onto". This characteristic ensures that for every possible output, there exists an input that will produce that output.

Proving surjectiveness often involves demonstrating that for every possible output value \(y\), you can find an \(x\) such that \(F(x) = y\). For example, the function \(F(x) = x^3 + 3x\) spreads across all real numbers, meaning any real number can be an output of this function given the right input.

Bijective functions are functions that are both injective and surjective. In this case, each element of the domain maps to a unique element of the codomain (injectivity), and every element of the codomain receives a mapping (surjectivity). This one-to-one and onto nature creates a perfect pairing between sets, much like a perfectly fitting set of keys to locks.

A bijective function guarantees that there is a one-to-one correspondence between elements of the domain and the codomain, paving the way for the function to have an inverse. In the original problem, the function \(F(x) = x^3 + 3x\) was shown to be bijective, which means it has a unique continuous inverse, affirming the solid relationship bijective functions have for defining a perfect match between sets.