Problem 2 - Algebra and Trigonometry

Question

Express the rule in function notation. For example, the rule square, then subtract 5 is expressed as the function \(f(x)=x^{2}-5 .\) Divide by \(7,\) then subtract 4

Step-by-Step Solution

Answer

The function is \(f(x) = \frac{x}{7} - 4\).

Step 1: Identify the operation

The given rule is "Divide by \(7\), then subtract \(4\)." The operation requires two steps: first, dividing a number \(x\) by \(7\), and second, subtracting \(4\) from the result.

Step 2: Express division in function notation

The operation "Divide by \(7\)" can be expressed as \(\frac{x}{7}\). This represents the first part of our function.

Step 3: Incorporate subtraction into the function

To integrate subtraction, we take the result from Step 2 and subtract \(4\). This gives us the complete function: \(f(x) = \frac{x}{7} - 4\).

Step 4: Write the function in standard notation

Combine all expressions into one function to fully express the rule. The final function is expressed in its standard form as follows: \(f(x) = \frac{x}{7} - 4\).

Key Concepts

Algebra and Its Role in Functions

Algebra is a cornerstone of mathematics that deals with symbols and the rules for manipulating these symbols. It's what allows us to create formulas and functions that model real-world problems. In this exercise, algebra is used to manipulate the symbol, known as a variable, to show relationships or transformations. For example, if we start with a variable represented by \( x \), algebra helps us mathematically express the series of operations or transformations applied to \( x \).

Algebra allows us to:

• Express mathematical rules in a concise form
• Use symbols to represent numbers in calculations
• Simplify complex formulas and expressions

In the given exercise, algebra helps us clearly define the order of operations: first, divide by \( 7 \), then subtract \( 4 \). This clear, methodical approach aids in avoiding confusion, especially when multiple operations need to be done in sequence.

Understanding Function Operations

Function operations involve performing algebraic operations on functions. Functions themselves are like machines that take an input, perform a specified series of operations on it, and then produce an output. In this context, function notation is used to define and express these operations clearly.

In function operations, you would typically:

• Identify the sequence of operations to be performed
• Apply each operation in the correct order
• Use function notation to write the operations concisely

For the exercise, the first operation was dividing by \( 7 \), denoted as \( \frac{x}{7} \). The second was subtracting \( 4 \). Combined, they form the complete function \( f(x) = \frac{x}{7} - 4 \), which indicates the operations to be performed on any input \( x \). This sequence respects the order of operations, which is crucial to obtaining the correct result.

Crafting Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators representing a specific value or set of values. In creating expressions, understanding how each aspect interacts is key to formulating accurate mathematical models.

Key elements of mathematical expressions include:

• Numbers: Constants that are either given or calculated
• Variables: Placeholders that can take various numerical values
• Operators: Symbols indicating operations such as addition, subtraction, multiplication, etc.

In this exercise, the expression \( \frac{x}{7} - 4 \) is crafted using these elements to represent the operations described. Here, \( \frac{x}{7} \) divides \( x \) by \( 7 \), and the \( -4 \) subtracts \( 4 \) from that result. By combining these components correctly, you create a mathematical expression that reflects the desired operations accurately, allowing anyone to follow the same instructions on any input \( x \).