Problem 34 - 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. $$ y=\sqrt{x-5}+1 $$

Step-by-Step Solution

Answer

Question: List the algebraic operations in the given function and find the domain based on the restrictions each operation places on the domain. Function: $$y = \sqrt{x-5} + 1$$. Answer: The algebraic operations in the function are subtraction, square root, and addition. The only restriction on the domain comes from the square root operation, which states that $(x-5) \ge 0$. Thus, the domain of the function is all real numbers greater than or equal to 5, written in interval notation as $$[5, \infty)$$.

Step 1: (a) Algebraic Operations and Restrictions

The function can be analyzed as follows: 1. We have subtraction operation: $$(x-5)$$. There is no restriction on the domain for subtraction. All real numbers are allowed. 2. We have square root operation: $$\sqrt{x-5}$$. In this case, we have a restriction, as the square root of a negative number is not a real value. Therefore, the argument of the square root must be greater than or equal to 0: $$(x-5) \ge 0$$. 3. We have addition operation: $$(\sqrt{x-5} + 1)$$. There is no restriction on the domain for addition. All real numbers are allowed. Based on these algebraic operations, the only restriction on the domain is due to the square root operation, and it is given by: $$(x-5) \ge 0$$.

Step 2: (b) Domain of the Function

To find the domain of the function, we must consider the restriction we found for the square root operation. The inequality expressing the restriction is: $$(x-5) \ge 0$$. Solving the inequality for x, we get $$x \ge 5$$. This means that the domain of the function is all real numbers greater than or equal to 5. In interval notation, the domain of the function is $$[5, \infty)$$.

Key Concepts

Algebraic Operations

In mathematics, especially when dealing with functions, understanding algebraic operations is essential. These operations include basics such as addition, subtraction, multiplication, and division. In some cases, operations might also include functions like square roots.

When evaluating functions, listing these operations in order can help identify any constraints on the domain. In our function, we have three main steps:

• First, the subtraction operation: \((x-5)\). Subtraction does not affect the domain. It accepts all real numbers.
• Second, we apply the square root operation: \(\sqrt{x-5}\). This is where domain restrictions become critical.
• Finally, there is an addition operation: \((\sqrt{x-5}+1)\), which doesn’t impose any restrictions either.

Identifying how each operation affects the function is crucial for determining the valid input values (the domain). Understanding these algebraic operations and their order aids in grasping more complex mathematical concepts.

Square Root Function

A square root function is a type of radical function characterized by the square root of a variable or an expression. It follows a predictable pattern and imposes certain limitations on its domain due to the nature of square roots.

The key property of square roots is that the radicand (the term under the square root) must be non-negative. This is because real numbers cannot have a square root for negative values, as they do not exist in the set of real numbers.

• For the function \( \sqrt{x-5} \), the restriction is that \((x-5)\) must be greater than or equal to zero.
• This leads us to solve the inequality \(x-5 \ge 0\), giving \(x \ge 5\) as the condition for valid input values.

Understanding square root functions and their restrictions helps in solving real-world problems where negative values wouldn't make sense, such as lengths and distances.

Restricting Domains

The concept of restricting domains in functions is fundamental to accurately determining which inputs will yield real number outputs. Some mathematical operations naturally impose these restrictions to maintain the realm of real numbers.

In our current discussion, it's the square root function that restricts the domain. As previously noted, the expression \(\sqrt{x-5}\) demands that the radicand \((x-5)\) be non-negative.

• This requirement limits the domain to \(x \ge 5\), meaning only values greater than or equal to 5 are valid.
• Hence, our final domain in interval notation: \([5, \infty)\).

Restricting domains is essential not just for solving algebraic problems but also in ensuring computations make sense within the context they are applied, like avoiding negative values in cost calculations or invalid times in physics problems.