Problem 35 - Algebra Form and Function

Question

For the functions, (a) List the algebraic operations in order of evaluation. What restrictions does each operation place on the domain of the function? (b) Give the function's domain. $$ y=4-(x-3)^{2} $$

Step-by-Step Solution

Answer

Answer: The domain of the function is all real numbers, represented as \(x \in \mathbb{R}\).

Step 1: Analyze function's structure

The structure of the function \(y = 4 - (x - 3)^2\) can be seen as following three sequence of operations: 1. Inside the parenthesis:\((x - 3)\) 2. The square operation: \((x - 3)^2\) 3. Subtraction operation: \(4 - (result)\)

Step 2: Determine restrictions to the domain

1. Inside the parenthesis: There are no restrictions on the input values of x in the first operation. So, no restriction from this step. 2. The square operation: Squaring any real number is possible, so there is no restriction on the input value from this operation. 3. Subtraction operation: Any real number can be subtracted from any other real number, so there are no restrictions from this step.

Step 3: Determine the function's domain

Since there are no restrictions on x values from any of the three operations, the function's domain is all real numbers. Domain: $$x \in \mathbb{R}$$

Key Concepts

Algebraic Operations

In mathematics, functions are often evaluated through a series of algebraic operations. These operations dictate the steps you need to follow to calculate the output of a function for any given input. When analyzing the function \( y = 4 - (x - 3)^2 \), we can break it down into a series of steps or operations that need to be performed in a particular order.

• First Operation: Inside the parenthesis \((x - 3)\).
• Second Operation: Squaring the result of the first operation, \((x - 3)^2\).
• Third Operation: Subtracting the squared result from 4.

Each operation directly affects the calculation of the function's value. Understanding each step's sequential execution is crucial for accurately determining the outcome. By familiarizing yourself with these algebraic manipulations, you learn how each step transitions the input to the output smoothly in the function's framework.

Domain Restrictions

Domain restrictions are critical in determining the possible inputs a function can accept. These restrictions arise from points where the function might be undefined or constrained by algebraic operations. In our example, we consider the function \( y = 4 - (x - 3)^2 \). Let's examine the operations again to see if any generate restrictions:

• Within the parenthesis, \((x-3)\) does not pose any restriction as subtraction can be performed on all real numbers.
• The square operation \((x - 3)^2\) allows any real input, as squaring real numbers presents no inherent restrictions.
• The subtraction \(4 - (result)\) is similarly unrestrictive, as any real number can be subtractively combined with another.

For this particular function, no operation limits the domain, hence, the function can accept all real numbers as valid inputs. In general, always check if operations like division (division by zero) or even roots (square root of a negative number) introduce exceptions to the domain.

Real Numbers

In the context of the function \( y = 4 - (x - 3)^2 \), the domain involves real numbers. Real numbers are all the numbers on the number line, encompassing both rational and irrational numbers. They are fundamental to the understanding and operations of functions.

• Rational Numbers: Include fractions and integers such as \( -3, 1, \frac{1}{2} \).
• Irrational Numbers: Numbers that cannot be expressed as fractions, such as \( \sqrt{3} \) or \( \pi \).

These encompass the possible inputs \( x \) for the function, reflecting its comprehensive domain. For algebraic functions like the one analyzed here, unless restricted by operations like division by zero or negative roots, the domain is typically the set of all real numbers: \( \mathbb{R} \). This ensures that every potential input has a corresponding output in the function’s value set.