Calculus — Mathematical Functions

A function is a unique relationship that can be made between numbers that enables us to relate one group of numbers to another group of numbers, or to create a group of numbers from another group using a specific relation.

Consider the list of counting numbers as the first group of numbers, group A, from 0, to 1, 2, 3, 4, 5, 6, … and so on. To each number in the first set of numbers, Set A, we are going to simply multiply by 2 and add 5 to create another set of numbers that will call Set B. This gives us the following table.

A function, a mathematical relationship between two sets of numbers, allows us to create on numerical set of data from another. Each element in the first set is related precisely to one element of the second set. In the case found in the table to the left, we have two columns: one to the left that we will call Set A, and one to the right called Set B.

We multiply the number in set A by two and add five, to arrive at its value in Set B on the right, $B = 2A + 5$. This is a mathematical function. We can write this out using familiar x and y variable notation as

\[ y = 2x + 5 \]

We can also express this as

\[ f(x) = 2x + 5 \]

Defining Functions

We define a function formally by saying that if we have two sets A and B, a set with elements that are ordered pairs (x, y) — where x is an element of A and y is an element of B — the function being a relation from A to B. This relationship between A and B defines the relationship between those two sets.

A function exists when each element of the first set related to exactly one element second set, and are used all the time in math, physics, and science to describe the behaviour of the relationship between two sets of numbers.

An element of first set is called the input, while the element of the second set is called output. For any function, if we know the input then the output is determined, since the function is a mathematical equation that allows us to compute the output. Thus we can say that the output is a function of the input. [1]

Basic Functions

The types of functions that most people will be familiar with, at least a few of them, come from geometry: the area of a square, the volume of a cube, or the area of a right-angle triangle. The next come from basic physics, like the velocity of a thrown ball, or the velocity of a fall due to gravity.

• The area of a square determined by its side length, given by the equation l^2 = A. The area is a function of the side length, with each designated as area (output) and side-length (input). [1]
• Velocity of ball thrown in the air described as function of the time that the ball is thrown in the air. This is given by the …
• Cost of mailing package is function of weight of package. [1]

Since functions have so many uses, it’s important to have precise definitions and terminology to study. [1] A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input exactly to one output. The set of inputs is called the domain, while the set of outputs defined by the function is called the range.

In addition to this, we have established a more formal language for, writing something like, $f(x) = x^2 + bx + c$ and saying something like “A function where f at x equals

Consider f where D ∈ R and the rule is to square the input. If we consider specifically the “function, f, at x=3″, the input is x=3 is assigned to the output y =3^2 = 9. Since every non-negative real number has a real-value square root, all non-negative numbers are elements of range of this function.

Since no real number exists with a negative square, negative real numbers not elements of range of this function. Conclusion that range is set of nonnegative real numbers.

In general, the practice is that when we refer to some function f with domain D, and often use x to denote input and y to denote associated output.

The variable x is the independent variable and y is the dependent variable. SInce y depends on x. Using function notation, we don’t use “y” but refer to the function directly. IN function notation we use y=f(x) stated “y equals f of x” and write a functon f(x)=x^2 for example.

[Figure 1.2, 1.3, and 1.4 … which I found to be more confusing and irrelevant than helpful.] [1, I think.]

Graphing Mathematical Functions

We can also visualize function plotting points (x,y) in the coordinate plane where y=f(x). The graph of a function is the set of all these points. Consider functuion f where domain is set D = {1,2,3} and the rule is f(x)=3-x.