Welcome to our exploration of function properties. A function is a fundamental mathematical concept that creates a relationship between two sets. For every input value in the domain, a function produces exactly one output value in the codomain. This unique mapping property is what distinguishes functions from general relations.
Two fundamental properties of functions are domain and range. The domain is the complete set of all possible input values that the function can accept. The range is the set of all actual output values that the function produces. For example, if we have f of x equals x squared with x between negative 2 and 2, the domain is the interval from negative 2 to 2, and the range is from 0 to 4.
Functions can be classified by their mapping properties. An injective or one-to-one function means that each output value corresponds to at most one input value. If f of x1 equals f of x2, then x1 must equal x2. A surjective or onto function means every element in the codomain is mapped to by at least one element in the domain. When a function is both injective and surjective, it is called bijective, creating a perfect one-to-one correspondence.