Question

Is a sequence a function? What is the difference between a sequence and the function f(x) with the same equation? Do the following problems to explore the difference. Let us consider the difference between t(n) = 2.3^3 and f(x) = 2.3^x

a. Is f(x) = 2.3^x a function? Why or why not

b. Is t(n) = 2.3^n a function? Why or why not

Answer 1

Both a sequence and a function pair inputs with outputs, but a sequence typically has a domain of natural numbers, creating discrete pairs, while a function often has the real numbers as its domain. Both t(n) = 2.3^n as a sequence and f(x) = 2.3^x as a function have unique outputs for their inputs, fulfilling the main criterion for being a function.

Step-by-step explanation:

Both a sequence and a function can be viewed as sets of ordered pairs, but they differ in certain aspects. A sequence is a function specifically where the domain is the set of natural numbers, and it outputs a list of numbers. In the case of t(n) = 2.3^n, we refer to the sequence as a function when n takes values in the natural numbers, providing a list of outputs based on this specific discrete domain. On the other hand, f(x) = 2.3^x is indeed a function as well, but its domain is usually considered to be all real numbers, which makes it continuous and defines it for any real number x.

Regarding whether the provided equations are functions:

1. f(x) = 2.3^x is a function because, for every value of x in the domain, there is exactly one corresponding output value. This respects the definition of a function, which states that each input should be associated with a single output.
2. t(n) = 2.3^n is also a function in the context of sequences because it assigns a single value of t for each natural number n.