Exploring Mathematical Concepts – Function Composition, the Number e, and Sequences
Question 1: Functions Composed Together
Composition of functions is an operation in math that combines two or more tasks to create a new function such that a value of one function derives a value in the other function. The functions are linked to each other in chains, meaning that they are applied successively (Aufmann & Nation, 2015). If two functions are f(x) and g(x), then their composition is denoted by (f ∘ g)(x), and by definition, it is f(g(x)). It is a composition where the function g(x) is first used to the input x, and then the resulting output is used as an input to function f(x). It is for that reason that there are several differences, primarily when the two are being merged either by composition or by product. Composition applies one function to the result of another, while a product of functions is akin to multiplying the results (Larson, 2016). In other words, the functions are composed sequentially—meaning the result of one goes to the input of the other—while the product of functions applies simultaneously, and the results are multiplied.
Question 2: The Number e
The numeral e equals approximately 2.7, which is an important mathematical constant often used in Calculus and exponential growth functions. It is known as the natural logarithm base or Euler number following its identification by the mathematician Leonhard Euler. The value of e is obtained from the natural logarithm and has close links with the area of exponential functions and the growth that is continuous. Some widely used mathematical and scientific relationships for which e is used are compound interests, radioactive decay, and electrical circuits, among others. Uses of e include many branches of mathematics, statistics, probabilities, and computer sciences, among others. Its precise measure is what is referred to as the irrational number, and this actually goes on to infinity, and the numbers do not have to repeat themselves in any way. The fact that e appears in numerous mathematical and scientific equations proves just how central it is and the relation it has to the foundation of the physical world.
Question 3: Arithmetic and Geometric Sequences
Arithmetic and geometric sequences are two sequence types that follow some patterns. The arithmetic and geometric sequences have a common difference or ratio between any two consecutive terms of a given sequence (Aufmann & Nation, 2015). The arithmetic sequence is one in which one adds the common difference to each term in a sequence to yield the successive term in this sequence. For example, in the arithmetic series 2, 5, 8, 11, 14, the common difference is three since it is added to the previous term to give the next. However, in the geometric series, the ratio is constant between each of them and is called the standard ratio, and when each term is multiplied by the ratio, it gives the next one of the series. Here, it is easy to determine that the geometric series has a ratio of 3 since each term is determined by taking the term before it and multiplying it by 3. The one key difference between the two sequence forms is how the terms are advanced. In arithmetic progress, it increases or decreases by a constant difference. In contrast, geometric progress increases or decreases by a constant factor, and thus, it is said to be an exponential progression.
References
Aufmann, R. N., & Nation, R. (2015). Algebra and trigonometry (8th ed). Belmont, CA: Cengage Learning.
Larson, R. (2016). Precalculus with limits (4th ed). Boston, MA: Cengage Learning.
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Question 
Think and Answer requires you to answer two questions. For each question, do not simply provide an answer; make sure you explain how you arrived at that answer. Even if your reasoning is wrong, you will still be credited for participation.
Exploring Mathematical Concepts – Function Composition, the Number e, and Sequences
Please write one or two paragraphs on each topic. Express your vision of the topic in your own words.
What is a composition of functions? What is the difference between the composition and the product of two functions?
Write a short paragraph about the number e.
What is common and what is different about arithmetic and geometric sequences?