Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$(4-2) !$$

Step-by-Step Solution

Answer

The value of \((4-2)!\) is 2.

Step 1: Understand the Expression

The given mathematical expression is \((4 - 2)!\). This consists of two parts: a subtraction and a factorial operation. First, we need to perform the subtraction inside the parentheses.

Step 2: Perform Subtraction

Inside the parentheses, we have \(4 - 2\). Calculate this subtraction to simplify the expression.\[ 4 - 2 = 2 \]

Step 3: Apply Factorial

After the subtraction, we have the expression \(2!\) remaining. The factorial of a number \(n\), denoted as \(n!\), is the product of all positive integers from 1 to \(n\).Calculate \(2!\):\[ 2! = 2 imes 1 = 2 \]

Step 4: Final Answer

The value of \((4 - 2)!\) is 2 after computing the subtraction and the factorial.

Key Concepts

Factorial

The factorial of a number is a fundamental concept in mathematics often denoted by an exclamation mark \(n!\).
Factorials are particularly useful in permutations, combinations, and probability calculations, among many other mathematical areas.
Here's how it works:

• The factorial of a positive integer \(n\) is the product of all positive integers less than or equal to \(n\).
• For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• The factorial of 0 is defined to be 1, i.e., \(0! = 1\).

In the exercise problem, we calculated \(2!\). This means we multiply every positive integer up to 2, which is \(2 \times 1 = 2\). So the factorial is a simple way of multiplying sequences of descending numbers.

Subtraction

Subtraction is one of the basic operations of arithmetic that involves finding the difference between numbers. When you subtract, you are essentially removing some quantity from another.
Key points about subtraction:

• It is the inverse of addition.
• The notation is straightforward: the symbol \(-\) is used, as in \(a - b\), which means subtract \(b\) from \(a\).
• In subtraction, \(4 - 2 = 2\) indicates that if you take 2 away from 4, you are left with 2.

Consider subtraction as taking steps backwards on a number line, where each step represents removing one unit from the total. In this exercise, subtraction had to be performed first since it was inside parentheses.

Mathematical Expression Evaluation

Evaluating a mathematical expression involves several steps to simplify and find an answer systematically. The goal is to perform all operations within the correct order and achieve a final result.
Here are some helpful tips:

• Identify and perform operations inside parentheses first.
• Proceed with multiplications, divisions, and further operations as per the hierarchy of operations, often remembered by "PEMDAS" (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
• Ensure that each step simplifies the expression while maintaining the correct mathematical relationships.

In our exercise, the subtraction was inside the parentheses, which needed addressing first. Then, the simplification of \(2!\) proceeded straightforwardly, demonstrating how recognizing and applying operations in the correct sequence lead to the solution.

Arithmetic Operations

Arithmetic operations are the foundation of mathematical calculations, involving addition, subtraction, multiplication, and division. Mastering these allows you to manipulate numbers easily and handle more complex mathematical problems.
Let's break them down:

• Additions are about combining values.
• Subtractions find differences between values.
• Multiplications are about repeated additions.
• Divisions distribute a number into equal parts.

Understanding the interplay of these operations is crucial, especially when expressions involve multiple operations requiring careful attention to the order of execution.
For example, the expression \( (4-2)!\) involves subtraction followed by multiplication for the factorial calculation, which illustrates the necessity of handling different operations with precision to reach an accurate outcome.