Functions are essential tools in describing real-world phenomena using mathematical language. They help articulate the relationship between varying quantities, playing a crucial role in calculus studies.
Concept of Function
The term “function” was introduced by German mathematician Leibniz (1646 – 1716) to express the dependency of one quantity on another. For instance, the area “A” of a square depends on its side “x” through the formula A = x², making A a function of x. Similarly, the volume “V” of a sphere depends on its radius “r” through the formula V = (4/3)πr³, making V a function of r.
A function is essentially a rule or correspondence that establishes a connection between two sets. In the context of the square and sphere examples, each square of a given side has a unique area, and each sphere of a given radius has a unique volume.
Exercise 1.1
Exercise 1.2
Exercise 1.3
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Exercise 1.4
Exercise 1.5
Definition (Function – Domain – Range)
Formally, a function “f” from a set X to a set Y is a rule or correspondence that assigns a unique element in Y to each element in X. The set X is referred to as the domain of f, and the set of corresponding elements in Y is the range of f. Unless specified otherwise, it is assumed that X and Y consist of real numbers.
Notation and Value of a Function
When a variable “y” depends on a variable “x” in a way that each value of x determines precisely one value of y, we say “y is a function of x.” Swiss mathematician Euler (1707-1783) introduced the symbolic representation y = f(x), read as “y is equal to f of x.” Functions are often denoted by letters such as f, g, h, F, G, H, etc.
A function can be conceptualized as a computing machine “f” that takes an input x, operates on it, and produces a unique output f(x). This output, denoted as f(x) or simply y, is the value of f at x, commonly referred to as the image of x under f. The variable x is the independent variable, while y is the dependent variable. In this context, we focus on functions where both variables are real numbers, specifically, a real-valued function of real numbers.