Problem 5 - Algebra and Trigonometry

Question

Express the function (or rule) in words. $$ f(x)=\frac{x-4}{3} $$

Step-by-Step Solution

Answer

Subtract 4 from \( x \) and divide by 3.

Step 1: Identify the Function Components

The given function is \( f(x) = \frac{x-4}{3} \). This function is a linear equation where \( x \) is the variable. The numerator of this fraction is \( x - 4 \), and the entire expression is divided by 3.

Step 2: Translate Mathematical Symbols into Words

The function \( f(x) = \frac{x-4}{3} \) can be expressed in words by describing each mathematical operation. The expression \( x - 4 \) indicates that you subtract 4 from \( x \), and then \( \frac{x-4}{3} \) shows that the result of \( x - 4 \) is divided by 3.

Step 3: Formulate the Rule in Words

Combine the translated components to express the function as a complete sentence: "To find the output of the function, subtract 4 from the input \( x \) and then divide the result by 3." This describes exactly what the function \( f(x) = \frac{x-4}{3} \) does to any input value \( x \).

Key Concepts

Function components

When we talk about function components, we refer to the different parts of a function that make up its structure. Functions are mathematical entities that connect an input, often denoted by \( x \), to an output, commonly represented as \( f(x) \). For the function \( f(x) = \frac{x-4}{3} \), there are several important components to recognize:

• Variable \( x \): This represents the input value. It's like the ingredient or starting point that we apply the function to.
• Operations: In this function, operations include subtraction and division. They show how the input is transformed to produce the output.
• Constant Terms: In this case, -4 and 3 are constants. They adjust the input in specific ways, affecting how the function behaves.

Understanding these components helps in grasping what the function \( f(x) = \frac{x-4}{3} \) does, as each part contributes to the calculation of the output values when different \( x \) values are plugged in.

Mathematical operations

Mathematical operations are the tools we use to manipulate numbers and variables to express relationships and calculate results. In the function \( f(x) = \frac{x-4}{3} \), two primary operations are used:

• Subtraction: The expression \( x - 4 \) indicates that we are subtracting 4 from any input value \( x \). This operation modifies \( x \) before any further steps are applied.
• Division: Once the subtraction is completed, the entire result \( x - 4 \) is divided by 3. This means we take the output of \( x - 4 \) and break it down into three parts, effectively determining one-third of that quantity.

Both operations are essential in defining the transformation process from an input value \( x \) to the output of the function \( f(x) \). By understanding and performing these operations in sequence, we can predict and calculate the result accurately.

Translating equations into words

Translating equations into words involves converting the symbolic representation of a math expression into a verbal description. This skill is crucial for understanding and communicating mathematical ideas effectively without needing to see the symbols.For the function \( f(x) = \frac{x-4}{3} \), we can translate it into a step-by-step verbal formula:

• First, take your input, which is usually represented by \( x \).
• Then, subtract 4 from this input value.
• After arriving at the subtraction result, divide that number by 3.

This can be summarized in a simple sentence: "To find the function's output, subtract 4 from the input \( x \) and then divide the result by 3." This translation helps us understand the function's procedure and intended operation, making it easier to work with and apply in various scenarios.