Problem 35 - Algebra 2

Question

REASONING Is the statement \(x_{n} \neq x_{n-1}\) sometimes, always, or never true if \(x_{n}=f\left(x_{n-1}\right) ?\) Explain.

Step-by-Step Solution

Answer

The statement is sometimes true, depending on the function.

Step 1: Understand the Function Notation

The expression \(x_n = f(x_{n-1})\) means that each term in the sequence is defined as the image of the previous term through the function \(f\). In other words, to find \(x_{n}\), you apply the function \(f\) to \(x_{n-1}\).

Step 2: Analyze the Identity Function Case

Consider the case where the function \(f(x) = x\). Substituting this into the formula gives \(x_n = x_{n-1}\). Therefore, in this case, the statement \(x_n eq x_{n-1}\) is never true. Every term in the sequence is the same as the previous one.

Step 3: Examine Non-Identity Function Cases

For other functions, such as \(f(x) = x + 1\), the function defines a sequence where each term is different from the previous one (e.g., \(x_n = x_{n-1} + 1\)). Therefore, \(x_n eq x_{n-1}\) is always true in this case. Likewise, more complex functions might generate terms that are different sometimes, but not always.

Step 4: Consider Functions that Cause Repetition

If \(f\) is a function that induces periodicity, some terms might equal a previous term. For instance, if \(f(x) = \cos x\), then \(x_n\) could eventually repeat values if the sequence includes values where \(x_{n} = x_{n-k}\) for some \(k\). This illustrates that \(x_n eq x_{n-1}\) can be sometimes true.

Step 5: Conclusion from Analysis

Upon reviewing the dependency of the result on the function \(f\), the statement \(x_n eq x_{n-1}\) is sometimes true, as it can depend on the particular form of \(f\). Some functions might always return different terms (or always the same), while others might only differentiably affect the sequence at specific instances.

Key Concepts

Function Notation

Function notation provides a convenient and clear way to express relationships between inputs and outputs. When you see the expression \(x_n = f(x_{n-1})\), this indicates a sequence where each term \(x_n\) can be computed by applying a function \(f\) to the previous term \(x_{n-1}\).

• Function notation allows us to succinctly describe how each term in a sequence is generated from the previous term.
• This notation is crucial in mathematics as it defines the rules or operations that transform one element into the next.

Understanding how to read and apply function notation is important because it lays the foundation for analyzing sequences and their behaviors across different types of functions.

Periodic Functions

A periodic function repeats its values at regular intervals, known as the period. Periodicity can have significant effects on sequences generated using such functions.

• If a sequence is defined using a periodic function, the terms may recur in a cycle.
• An example is the cosine function \(f(x) = \cos x\), which has a period of \(2\pi\).
• This periodic behavior means that after a certain number of terms, the sequence will begin repeating its prior values.

Sequences that exhibit periodicity showcase how understanding the function's properties guides predictions about the sequence behavior over time.

Identity Function

The identity function is a special kind of function where the output is always the same as the input, written as \(f(x) = x\). When this function is applied in a sequence, every term becomes identical to the previous one:

• For example, if \(f(x)\) is the identity function, then \(x_n = x_{n-1}\) for all \(n\).
• This means \(x_n eq x_{n-1}\) is never true, as no change occurs between terms.

Knowing the effects of an identity function is useful, particularly when determining how a sequence remains constant, contrary to sequences derived from more dynamic functions.

Sequence Analysis

Sequence analysis involves examining the behavior and characteristics of sequences generated by various functions. It helps in understanding whether terms of the sequence are equal, increase, decrease, or follow a specific pattern.

• Key to analyzing sequences is determining if \(x_n = x_{n-1}\) is true, false, or variable.
• For linear changes, the relationship between terms is consistent (e.g., \(f(x) = x + 1\) makes \(x_n = x_{n-1} + 1\), so terms always differ).
• When functions induce periodicity, terms might repeat nonlinearly, introducing variability as seen in periodic functions.

Overall, analyzing sequences requires a deep dive into the function's behavior to predict and describe the specific properties of the sequence in question.