Problem 1 - Precalculus

Question

Find an expression for \(f(x)\) and state its domain. \(f\) is a function that takes a real number \(x\) and performs the following three steps in the order given: (1) multiply by \(2 ;(2)\) add \(3 ;\) (3) divide by 4 .

Step-by-Step Solution

Answer

The function is \(f(x) = \frac{2x + 3}{4}\) with domain \( \mathbb{R} \).

Step 1: Function Representation

Start by expressing the function as a sequence of operations. We need to define each operation in the sequence that produces \(f(x)\). The sequence is: multiply by 2, add 3, and divide by 4.

Step 2: Apply the First Operation

Begin with the variable \(x\). The first operation is to multiply by 2, resulting in the expression \(2x\). This represents taking \(x\) and performing the first step: \(x \to 2x\).

Step 3: Apply the Second Operation

Take the result from the previous step, \(2x\), and apply the second operation: add 3. This gives us the new expression \(2x + 3\). This represents \(2x \to 2x + 3\).

Step 4: Apply the Third Operation

Finally, divide the result from the previous step, \(2x + 3\), by 4. This will result in the expression \(\frac{2x + 3}{4}\), which defines the complete transformation of \(x\) to \(f(x)\).

Step 5: Define the Function

Write the expression for \(f(x)\) based on the operations we performed: \[ f(x) = \frac{2x + 3}{4} \]. This is the complete expression for the function.

Step 6: Determine the Domain

Analyze the expression \(\frac{2x + 3}{4}\) to determine its domain. Since this is a linear expression divided by a constant (4), there are no restrictions; \(x\) can be any real number. Therefore, the domain of \(f(x)\) is all real numbers: \( \mathbb{R} \).

Key Concepts

Domain of a Function

In mathematics, the domain of a function is the complete set of possible input values (typically represented by the variable \(x\)) for which the function is defined. When determining the domain, we ask: Are there any values of \(x\) that do not work in the function? Or, are there any operations that might cause undefined results (such as division by zero or square roots of negative numbers)?
In the given problem, the function is expressed as \(f(x) = \frac{2x + 3}{4}\). This is a simple linear function divided by a constant. Essentially, it expresses a transformation that can handle any real number without causing any discrepancies. As there are no square roots or variables in the denominator that could lead to division by zero, there aren't any constraints based on mathematical exceptions.
Thus, the domain of this function is all real numbers, written as \( \mathbb{R} \). This means that no matter what real number you substitute for \(x\), the expression \(\frac{2x + 3}{4}\) is always perfectly valid and provides a definite outcome.

Operations in Functions

Functions often involve a variety of operations such as addition, subtraction, multiplication, division, and even more complex transformations. Understanding how to queue and execute these operations is essential in mathematics.
In this exercise, we're dealing with three sequential operations that form the expression for \(f(x)\):

• First, multiply \(x\) by 2 to get \(2x\).
• Next, take \(2x\) and add 3 to arrive at \(2x + 3\).
• Finally, divide the sum \(2x + 3\) by 4 to produce \(f(x) = \frac{2x + 3}{4}\).

These operations are carried out in a specific order, refining the result at each stage. The process will look like a transformation or a pipeline where each step modifies and flows from the previous one, culminating in the finalized function. This sequential approach is pivotal because changing the order could result in different outcomes or even errors that break the expression. By keeping the operations in the stipulated sequence, we ensure that the logical integrity of the function remains intact.

Linear Transformation

A linear transformation refers to a kind of mapping of functions whereby you apply linear operations (such as scaling and translation) to input values. It’s crucial because it describes how a function will transform its argument, and understanding it can be pivotal in solving more complex problems in fields like physics and engineering.
In the specific case of the function expression \(f(x) = \frac{2x + 3}{4}\), each step is part of a linear transformation process. The multiplication by 2 scales the function, enlarging each input value. Adding 3 then shifts the entire outcome vertically by a constant, reflecting that the output has been translated inside the expression. Finally, dividing by 4 scales the result again, bringing the expression down to an average out.Thus, the entire formula represents a linear transformation of \(x\) into \(f(x)\). Each step animates how a straightforward variable undergoes transformations into a different representation of itself, bound by linear characteristics that maintain proportionality and additivity throughout. This concept is foundational in understanding how one can map or alter functions while maintaining the algebraic integrity of linear relationships.