Question

Suppose \(A=\\{a, b, c, d\\}, B=\\{2,3,4,5,6\\}\) and \(f=\\{(a, 2),(b, 3),(c, 4),(d, 5)\\} .\) State the domain and range of \(f\). Find \(f(b)\) and \(f(d)\).

Short answer

The domain of function f is \{a, b, c, d\}, the range is \{2, 3, 4, 5\}. The output f(b) is 3 and f(d) is 5.

Step-by-step solution

Identify Domain of Function f

The domain of a function is the set of all possible inputs for the function. In this case, the domain is set A, which means the domain of function f is \{a, b, c, d\}.

Identify Range of Function f

The range of a function is the set of all possible outputs of the function. In this case, not all elements in set B are used in function f, so the range of f is the set of all second elements in the ordered pairs of set f, which are \{2, 3, 4, 5\}.

Find f(b)

The output of the function for input 'b' is the second element of the ordered pair in function f where the first element is 'b'. Looking at set f, we can see the ordered pair with 'b' is (b, 3), so f(b) = 3.

Find f(d)

Similarly, the output of the function for input 'd' is the second element of the ordered pair in function f where the first element is 'd'. Looking at set f, we can see the ordered pair with 'd' is (d, 5), so f(d) = 5.

Key concepts

Set Theory

At the heart of understanding mathematics, particularly when it comes to functions, is set theory. It serves as the foundation of mathematical logic and helps define the very notion of a function. In simple terms, set theory deals with the study of well-defined collections of objects, which we call 'sets.' Each object in a set is unique and is referred to as an 'element'. Sets are usually denoted by capital letters, such as A, B, C, and so on.

Sets can be composed of anything—numbers, letters, symbols, or even other sets—and they are fundamental in defining the domain and range of a function, just like in the given exercise where set A and B are used. Interestingly, set theory is so versatile that it finds application in various branches of mathematics including logic, combinatorics, probability, and more. It's crucial to grasp the basics of set theory to understand other advanced mathematical concepts.

Functions in Mathematics

A function is a special relationship between values: each of its input values gives back exactly one output value. Imagine functions as a 'machine' or a 'black box' that takes an input, does something to it, and then gives the exact output. If we have a set of inputs and a set of permissible outputs, a function defines a specific pairing between these two sets.

In mathematical terms, a function is often represented by f(x), where 'f' stands for function and 'x' is the input value. The set of all possible input values is called the domain, and the set of possible outputs is known as the range. Functions are fundamental in mathematics because they can model real-world phenomena in a concise way, allowing us to make predictions and understand relationships between variables. Our example function f pairs elements from set A (domain) to elements in set B (range), thus creating ordered pairs that show the input-output relationship.

Ordered Pairs

Understanding Ordered Pairs

Ordered pairs are a way to represent the relationship between two objects, typically denoted as (x, y) where 'x' is the first element and 'y' is the second. The order of elements is crucial—(a, b) is not the same as (b, a) unless a equals b. In the context of functions, each input (from the domain) is associated with an output (from the range) and this association is represented using ordered pairs.

The beauty of ordered pairs is that they provide a clear visual understanding of the function's pairing process. This concept is not only central in coordinate plane graphing but also in defining functions. In the exercise, for example, the ordered pair (a, 2) tells you that the function f pairs the element 'a' from the domain with '2' from the range. Grasping this concept aids in recognizing patterns and relationships within the function.