Definition 1.1 (Function): A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain* of

Here are the definitions and remarks from the provided text: Definition 1.1 (Function): A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain of the function, while the set of outputs is known as the range* of the function. Definition 1.2 (Function of a Real Variable): Let N R. A function f of a real variable is a rule or law which assigns to each x N exactly one y R. The variable x is called the argument or the independent variable, and the variable y is termed the dependent variable*. Definition 1.3 (Domain, Range and Co-domain): The set N is known as the domain of function f and is denoted by D(f) or Dom(f). Similarly, a set \y = f(x) : x D(f)\ is called the range of f, denoted by R(f). The general term for function is called mapping. Range is a subset of the co-domain*. Remark 1.1: 1. The domain is a part of the definition of the function. If the domain is not specified, we take it as the natural domain, which is the largest possible domain. 2. Two functions f and g are equal, that is, f = g, if the following two conditions are satisfied: D(f) = D(g) (they have the same domain). f(x) = g(x) for all x in their common domain. Send me the next one 📸