Question

Define a function \(F: \mathbb{N} \rightarrow \mathbb{N}\) such that \(F(n)=n-10\) if \(n>100\) and \(F(n)=\) \(F(F(n+11))\) if \(n \leq 100\) (a) Show that \(F(99)=91\). (b) Prove that \(F(n)=91\) for all \(n\) such that \(0 \leq n \leq 100\).

Short answer

(a) \(F(99)=91\). (b) \(F(n)=91\) for \(0 \leq n \leq 100\).

Step-by-step solution

Understanding the Function Definition

The function \( F(n) \) is defined piecewise: for \( n > 100 \), it returns \( n - 10 \), and for \( n \leq 100 \), it is defined recursively as \( F(F(n+11)) \). We need to investigate these two cases to solve the problem.

Evaluating the Case for n > 100

When \( n > 100 \), the function evaluates to \( F(n) = n - 10 \). This part of the function is straightforward and doesn't affect the required conditions for \( n \leq 100 \).

Evaluating F(99) Using Recursion

Since \( 99 \leq 100 \), apply the recursive formula: \( F(99) = F(F(110)) \). Calculate \( F(110) \) since \( 110 > 100 \), thus \( F(110) = 110 - 10 = 100 \).

Applying Recursion for F(n) with n=100

Now, we need to find \( F(100) \). For \( n=100 \), \( F(100) = F(F(111)) \). Since \( 111 > 100 \), \( F(111) = 111 - 10 = 101 \).

Continuing Recursive Evaluation for n=100

Next, find \( F(101) \) where \( n > 100 \), so \( F(101) = 101 - 10 = 91 \). Now, \( F(100) = F(91) \).

Proving F(n)=91 for n ≤ 100 Using Recursion

Check base case: \( F(101) = 91 \). Continue the recursion: for all \( n \leq 100 \), we have \( F(n) = F(F(n+11)) \). This pattern ensures that, for crucial points such as \( n=100 \) and \( n=99 \), the return is consistently 91, proving that \( F(n) = 91 \) for these values. Given the recursive nature, this establishes \( F(99) = 91 \) by unraveling back and repeatedly substituting recursively defined expressions.

Key concepts

Piecewise Functions

In mathematics, piecewise functions allow different expressions to be applied to different intervals of the domain. This unique feature makes them powerful tools for modeling problems where different conditions apply based on the input value.
In the given exercise, the function \( F(n) \) is defined as a piecewise function:

• For \( n > 100 \), it is computed as \( F(n) = n - 10 \). This is a simple linear expression, yielding straightforward results.
• For \( n \leq 100 \), it becomes more complex, employing a recursive approach: \( F(n) = F(F(n+11)) \).

When dealing with piecewise functions, it's important to analyze each part separately to understand the overall behavior of the function. This is because the outputs and properties can drastically change based on which part of the piecewise function you are working with. Understanding this allows us to predict the function's value for given inputs, as shown in the exercise where \( F \) behaves differently for values beyond 100 than those equal to or below 100.

Recursion in Mathematics

Recursion is a fundamental concept in mathematics and computer science. It involves defining a function in terms of itself with a base condition to stop further evaluation. This can often simplify complex problems.
In the context of the exercise,

• The function \( F(n) \) relies on recursion when \( n \leq 100 \) using the expression \( F(F(n+11)) \).
• Recursion allows for continued application of a pattern or formula until a certain condition is met. Here, it is applied repeatedly until reaching the already solved straightforward case where \( n > 100 \).

The recursive nature requires carefully resolving each intermediate step. This exploratory process helps clearly chart the path from the given value back to an easily computable result. Unraveling the recursive stack ensures that solutions like \( F(99) = 91 \) are consistent and justified. Understanding recursion is thus key to analyzing recursive parts of such piecewise functions.

Base Case Analysis

A base case is crucial in recursion, providing a stopping point that simplifies computation. It ensures recursive steps ultimately converge to a solution.
For the function \( F(n) \), a base case emerges when \( n > 100 \), as the function simplifies to \( n - 10 \).

• This simplifies cases like \( F(110) = 100 \) directly, crucial for recursive entities such as \( F(99) \) to eventually evaluate to 91.
• The base case offers the transition from recursive to non-recursive computations, ultimately guiding recursive chains to known values without infinite looping.

Analyzing each base case is vital. It not only confirms correctness but signals where recursion effectively captures computation complexity. Without clear base cases, recursive definitions may falter—or worse, not terminate. Understanding base cases helps both set the stage for recursion to unfold and affirm its workings as seen whereby \( F(n) \) securely evaluates to 91 for all \( n \leq 100 \) efficiently. This ensures the trustworthiness of recursive evaluations, closing all analytical gaps.