Problem 7 - Algebra and Trigonometry

Question

Express the rule in function notation. (For example, the rule "square, then subtract 5\("\) is expressed as the function \(f(x)=x^{2}-5 .\) Subtract \(5,\) then square

Step-by-Step Solution

Answer

\(f(x) = (x - 5)^2\)

Step 1: Understand the Rule

The given rule asks us to subtract 5 from a number and then square the result.

Step 2: Express the First Operation

Subtract 5 from a variable, say \(x\). This can be represented as \(x - 5\).

Step 3: Apply the Second Operation

Square the result from the first operation. This will be expressed as \((x - 5)^2\).

Step 4: Write in Function Notation

Represent the operation using function notation. We express this as \(f(x) = (x - 5)^2\). This function defines the operation of subtracting 5 from \(x\) and then squaring the result.

Key Concepts

Algebra

Algebra serves as the foundation of higher mathematics, dealing with symbols and the rules for manipulating these symbols. It allows us to create general representations of relationships and patterns using variables like \(x\), \(y\), etc. In the original exercise, we use algebra to express a relationship in function notation.
By handling variables, we can describe operations that apply to any number, offering flexibility in problem-solving. This versatility allows us to analyze and predict outcomes in varied scenarios.

• Variables represent numbers or values that can change.
• Algebraic expressions like \(x - 5\) show relationships between numbers.
• By simplifying expressions, we can solve equations and find unknowns.

Through this algebraic method, we can effectively record and solve equations that model real-world phenomena.

Mathematical Operations

Mathematical operations are fundamental processes in mathematics, including addition, subtraction, multiplication, and division. Each operation performs a specific task on numbers or expressions.
The problem at hand involves two primary operations: subtraction and squaring.
1. **Subtraction**: This operation decreases a value by a specific amount. Here, we subtract 5 from a variable \(x\), which gives us \(x - 5\). This step is vital because it changes the starting value to a smaller one.2. **Squaring**: Squaring a number means multiplying it by itself, written as \((x-5)^2\). This operation significantly increases the modified value by altering its scale.
Understanding and combining these operations is crucial in developing accurate mathematical expressions. Each operation follows specific rules that need to be applied in the correct order, known as the order of operations.

Function Representation

Function representation in mathematics is a way to describe a unique relationship between inputs and outputs using a specific notation. This notation, called function notation, is represented as \(f(x)\), where \(f\) names the function and \(x\) is the input variable.
In function notation, \(f(x) = (x - 5)^2\), the input \(x\) undergoes operations specified by the function's rule. Here, the operations involve subtracting 5 and then squaring the result.
Functions allow us to:

• Model real-world scenarios systematically, indicating a direct relationship between two quantities.
• Predict outcomes for different inputs, providing a clear depiction of how changes affect results.
• Simplify complex problems by breaking them into manageable steps.

Function notation is essential for communicating mathematical ideas efficiently, offering a consistent way to express operations and their results.