Problem 33 - 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. $$ y=\frac{2}{x-3} $$

Step-by-Step Solution

Answer

Answer: The algebraic operations in order of evaluation are: 1. Subtraction: \(x-3\) 2. Division: \(\frac{2}{x-3}\) Restrictions on the domain of each operation are: 1. Subtraction has no domain restrictions. 2. Division has domain restriction: \(x \neq 3\) The domain of the function is: $$ Domain = \{x \in \mathbb{R} \ | \ x \neq 3\} $$ or, in interval notation: $ Domain = (-\infty, 3) \cup (3, \infty) $.

Step 1: Identify the algebraic operations in order of evaluation

For the given function $$y = \frac{2}{x-3},$$ we can see that there are two main algebraic operations: subtraction and division. The operations should be evaluated in the following order: 1. Subtraction: \(x-3\) 2. Division: \(\frac{2}{x-3}\)

Step 2: Identify the restrictions each operation places on the domain

Let's now consider the restrictions placed on the domain by each operation: 1. Subtraction: There are no restrictions on the domain for subtraction. All real numbers are allowed. 2. Division: Since we cannot divide by zero, the value of the denominator \((x-3)\) cannot be zero. Therefore, the restriction is that \(x \neq 3\).

Step 3: Determine the domain of the function

Combining the restrictions from the individual operations, we find that the only restriction on the domain of the function is that \(x \neq 3\). Therefore, the domain of the function is all real numbers except 3, which can be represented as: $$ Domain = \{x \in \mathbb{R} \ | \ x \neq 3\} $$ or, in interval notation: $ Domain = (-\infty, 3) \cup (3, \infty) $.

Key Concepts

Algebraic Operations

In the study of mathematics, algebraic operations refer to the basic mathematical processes that include addition, subtraction, multiplication, and division. When dealing with functions, these operations might influence how we calculate function values or determine a function's domain.

Consider the function given by \( y = \frac{2}{x-3} \). There are two algebraic operations involved here:

• Subtraction \((x - 3)\): This operation subtracts 3 from \(x\), which poses no restrictions on \(x\) since subtraction is defined for all real numbers.
• Division \(\left(\frac{2}{x-3}\right)\): This operation divides 2 by \((x - 3)\). Division by zero is undefined in mathematics, so for this function, \(x\) cannot be 3 since that would make the denominator zero.

Understanding these operations helps us see why certain values cannot be included in the function's domain. It sets the groundwork for determining where the function is applicable.

Function Domain

The domain of a function refers to the complete set of all possible input values \(x\) that the function can accept without encountering mathematical impossibilities such as division by zero. It's essential to analyze each algebraic operation within a function to determine these permittable values.

For the function \( y = \frac{2}{x-3} \):

• The subtraction \((x-3)\) applies to all real numbers, meaning it doesn't restrict the domain.
• The division means we must exclude any value of \(x\) that causes division by zero. Here, \(x = 3\) makes the denominator zero, leading to an undefined expression.

Therefore, the domain of this function includes all real numbers except 3. This is typically written as: \[Domain = \{x \in \mathbb{R} \ | \ x eq 3\}\]Or in interval notation: \[Domain = (-\infty, 3) \cup (3, \infty)\]Thus, while subtraction itself does not change the potential inputs, division requires careful consideration of values that result in zero in the denominator.

Restrictions on Domain

Restrictions on a function's domain arise when certain operations within the function are not defined for specific inputs. This can occur primarily due to division by zero, negative square roots (for real numbers), or undefined logarithms of non-positive numbers. Understanding these restrictions is crucial for correctly defining a function's domain.

In \( y = \frac{2}{x-3} \), there is a restriction due to division. Key points include:

• The division by \(x - 3\) implies \(x\) cannot be 3. If \(x = 3\), the expression \(x - 3\) becomes zero, causing the division to be undefined.

Thus, to find the allowable domain:

• Determine what values make the denominator zero and exclude them from the domain.

For our function, with \(x eq 3\), the domain is all real numbers except 3. Recognizing these restrictions ensures that we only include permissible input values, avoiding mathematical errors and maintaining the function's integrity.