Problem 38 - 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. $$ y=\frac{7}{4-(x-3)^{1 / 2}} $$

Step-by-Step Solution

Answer

Question: List the algebraic operations in order of evaluation and identify the restrictions that each operation places on the domain of the function $$y = \frac{7}{4 - (x - 3)^{1/2}}$$, and determine the domain of the function. Answer: The algebraic operations with restrictions on the domain are the square root: $$(x-3)^{1/2}$$, which requires $$x \geq 3$$, and the division: $$\frac{7}{4 - (x - 3)^{1/2}}$$, which requires $$x \neq 19$$. The domain of the function is $$\{x \in \mathbb{R} | x \geq 3, x \neq 19\}$$, or equivalently, $$[3, 19) \cup (19, \infty)$$.

Step 1: (a) Identify algebraic operations and their restrictions on the domain

There are two important algebraic operations in the function $$y = \frac{7}{4 - (x - 3)^{1/2}}$$ that have an impact on the domain: 1. Square root: $$(x-3)^{1/2}$$ The square root function requires a non-negative value inside the square root, which means: $$x - 3 \geq 0$$ Hence, the restriction for the square root operation is: $$x \geq 3$$ 2. Division: $$\frac{7}{4 - (x - 3)^{1/2}}$$ The division requires that the denominator should not be equal to zero. So, we need to ensure: $$4 - (x - 3)^{1/2} \neq 0$$ $$ (x - 3)^{1/2} \neq 4$$ This implies, the restriction for the division is: $$x \neq 19$$ since $$(19 - 3)^{1/2} = 4$$

Step 2: (b) Determine the domain of the function

By combining the restrictions found in part (a), we can determine the overall domain of the function: - The square root operation requires $$x \geq 3$$ - The division operation requires $$x \neq 19$$ Therefore, the domain of the function can be represented using set notation or interval notation: Set notation: $$\{x \in \mathbb{R} | x \geq 3, x \neq 19\}$$ Interval notation: $$[3, 19) \cup (19, \infty)$$

Key Concepts

Algebraic Operations

Understanding algebraic operations in a function can clarify restrictions on the domain. For the given function, we will explore how these operations affect the permissible values of the variable:

• **Exponentiation**: Since there is a square root, the expression \((x-3)^{1/2}\) involves an implicit exponentiation operation, where \(x\) needs certain values to maintain a valid function.
• **Subtraction**: The operation \(x-3\) sets the stage for what follows.
• **Square Root**: Extracting the square root \((x-3)^{1/2}\) means the expression inside should be non-negative. Therefore, \(x-3 \geq 0\), translating to \(x \geq 3\), restricting the domain.
• **Subtraction Again**: From \(4 - (x-3)^{1/2}\), this operation continues to affect the outcome and the validity of the domain.
• **Division**: The division by \((4 - (x-3)^{1/2})\) implies the denominator cannot be zero. Therefore, \(4 - (x-3)^{1/2} eq 0\) becomes a critical condition.

This step-by-step consideration of operations outlines why the domain of a function must be carefully analyzed to ensure all operations remain valid.

Square Root Function

A square root function is a fundamental mathematical element with principles that influence the domain it can accept. When evaluating a square root such as \((x-3)^{1/2}\), it's crucial to consider the following:

• **Non-Negativity**: The expression inside the square root must be zero or positive. For \((x-3)\), this means \(x-3\geq 0\). Simplifying gives \(x \geq 3\).
• **Impact on Domain**: Since a square root demands specific conditions to produce real numbers, it automatically restricts the values \(x\) can take.
• **Equality and Beyond**: When thinking about equations, the square root affects conditions wherein equality and inequality must be defined, such as not allowing the result to exceed a certain value subordinated by division operations.

The influence of the square root is significant, as it precisely dictates the starting boundary for the domain through these simple yet powerful criteria.

Division in Functions

In functions involving division, one must always ensure the denominator does not become zero.
In \( y = \frac{7}{4-(x-3)^{1/2}} \), division places unique restrictions:

• **Non-Zero Denominator**: For any division, the denominator must never be zero. Here, \(4 - (x-3)^{1/2} eq 0\) is the key condition.
• **Relationship with Square Root**: Since this denominator operation relates directly to the square root, it extends the impact of both operations. Solving \( (x-3)^{1/2} = 4 \) gives \(x = 19\) as a value to exclude.
• **Combined Conditions**: Integrating the condition with the square root's domain conditions (\(x \geq 3\)), we get a refined eligible domain. \(x eq 19\) complements this by subtracting the impossible result from acceptable values.

By managing division along with other operations, students ensure validity in functional expressions, forming precise comprehensions about where a function holds true.