Problem 3 - 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) divide by \(4 ;(2)\) add \(3 ;\) (3) multiply by 2 .

Step-by-Step Solution

Answer

The expression for \(f(x)\) is \(\frac{x}{2} + 6\); its domain is all real numbers, \(\mathbb{R}\).

Step 1: Understand the process

We are given a function process that acts on a real number \(x\) through three operations. We start by identifying these operations: dividing by 4, adding 3, and then multiplying by 2.

Step 2: Perform the first operation

First, divide the input \(x\) by 4. This can be represented as \(\frac{x}{4}\).

Step 3: Perform the second operation

Take the result from the first operation, \(\frac{x}{4}\), and add 3 to it. This becomes \(\frac{x}{4} + 3\).

Step 4: Perform the third operation

Multiply the result from the second operation \(\left(\frac{x}{4} + 3\right)\) by 2. This gives us \(2 \times \left(\frac{x}{4} + 3\right)\).

Step 5: Simplify the final expression

Expand the expression from Step 4 to get the final expression for \(f(x)\). This results in \(f(x) = 2\left(\frac{x}{4}\right) + 2 \times 3 = \frac{x}{2} + 6\).

Step 6: Determine the domain

Since the original input \(x\) can be any real number and noting that none of the operations limit the input (division by 4 is fine as long as it's not zero, which isn't a restriction on \(x\)), the domain of the function \(f(x)\) is all real numbers, \(\mathbb{R}\).

Key Concepts

Real Numbers

Real numbers are a significant concept in mathematics, encompassing all numbers that can be found on the number line. This includes positive numbers, negative numbers, zero, fractions, and irrational numbers like \sqrt{2}\. These numbers are crucial in various mathematical operations and processes. In the context of our exercise, the function \(f(x)\) takes a real number \(x\) and performs a series of operations. Since \(x\) can be any real number, it means we have a vast set of values that \(x\) can potentially be, ranging from negative infinity to positive infinity.
A few important properties of real numbers include:

• Closed under addition, subtraction, and multiplication. This means when you add, subtract, or multiply any two real numbers, the result is also a real number.
• Dividing by a non-zero real number also results in a real number.
• Real numbers can be positive, negative, or zero.
• They have no gaps between them on the number line; every point on the line corresponds to a real number.

Domain of a Function

The domain of a function refers to the complete set of possible input values (\(x\) values) for which the function is defined. For the function \(f(x) = \frac{x}{2} + 6\), we need to determine what values \(x\) can take. This solution shows how each operation in our function process does not restrict \(x\).

Here are some points to consider when determining the domain of a function:

• Identify any operations in the function that might impose restrictions, such as division by zero or taking the square root of a negative number.
• If none of these operations are present on the input \(x\), like in our exercise, then the domain is simply all real numbers, \(\mathbb{R}\).

The function \(f(x)\) takes any real number \(x\), because operations such as division by 4 or addition/subtraction of constants do not exclude any real numbers. Thus, the domain of \(f(x)\) is all real numbers.

Arithmetic Operations

Arithmetic operations include basic mathematical processes such as addition, subtraction, multiplication, and division. These operations are the building blocks of more complex mathematical expressions and functions, like the one seen in this exercise. For \(f(x)\), we utilized these operations in a specific sequence to transform the input \(x\) into a new value.

In this exercise, each step of the function involves an arithmetic operation:

• **Division:** Dividing \(x\) by 4. This reduces \(x\) and results in \(\frac{x}{4}\).
• **Addition:** Adding 3 to the result. This transforms \(\frac{x}{4}\) into \(\frac{x}{4} + 3\).
• **Multiplication:** Finally, multiplying the sum by 2, which scales the result as \(2 \times \left(\frac{x}{4} + 3\right)\).

It’s important to follow the operations in the specified order, as changes in order can lead to different results. In function composition, each operation builds off the previous result, making order crucial for accuracy in the outcome. Familiarity with these operations is essential for manipulating and solving expressions efficiently.