Problem 8 - Algebra and Trigonometry

Question

Express the function (or rule) in words. $$ k(x)=\sqrt{x+2} $$

Step-by-Step Solution

Answer

The function takes a number, adds 2, then calculates the square root.

Step 1: Understand the Function

The exercise involves expressing the mathematical function \(k(x) = \sqrt{x+2}\) in words. This function takes an input \(x\) and applies a specific operation to it.

Step 2: Break Down the Function Components

The function \(k(x) = \sqrt{x+2}\) consists of two main components: Adding 2 to the input \(x\) and taking the square root of the result. This gives insight into the operations needed to express the rule in words.

Step 3: Formulate the Verbal Expression

To express the function in words, start with the input \(x\). The function adds 2 to this input. Then define that the next operation involves computing the square root of the addition result. Putting this together provides a complete verbal rule.

Key Concepts

Mathematical Operations

Mathematical operations form the backbone of any function in mathematics. They include a range of actions like addition, subtraction, multiplication, and division. When working with functions, these operations transform inputs into outputs by applying these actions sequentially or simultaneously. In the function \( k(x) = \sqrt{x+2} \), understanding the operations involved is key to comprehending how the function behaves. Breaking Down Operations:

• Addition: The function specifies adding 2 to the input \( x \). This means if you were given an input value, you would first increase it by 2.
• Square Root: After adding 2, the next operation is to apply the square root to the resulting value.

Breaking these operations down into simple steps helps in easily translating them into words and expressing the function clearly.

Square Root Function

The square root function, often depicted as \( \sqrt{x} \), is a core mathematical concept. It represents the operation of finding a number which, when multiplied by itself, gives the original number. The function \( k(x) = \sqrt{x+2} \) specifically involves taking the square root of the sum of \( x \) and 2.Key Points about Square Roots:

• Definition: The square root function reverses the squaring process. If \( y^2 = x \), then \( y = \sqrt{x} \).
• Domain: The values inside the square root must be non-negative as you cannot take the square root of a negative number within the standard real number system.
• Effect: Taking the square root generally results in a smaller number, closer to zero, particularly if the input is larger.

Recognizing square root functions' behavior and constraints is essential in discussing or interpreting any function that includes it.

Verbal Expression of Functions

Expressing functions in words is a valuable skill that allows you to communicate mathematical rules clearly and effectively. For the function \( k(x) = \sqrt{x+2} \), converting mathematical symbols into a verbal expression simplifies understanding for many students.Steps to Verbalize a Function:

• Identify Components: Determine the operations involved—as we did in previous sections with addition and square root.
• Sequence the Operations: Outline the order in which these operations occur.
• Express in Sentence: For \( k(x) \), you could say: "Take the input \( x \), add 2, and then compute the square root of that sum."

This verbalization makes the function accessible, ensuring that anyone, regardless of their mathematical prowess, can grasp the process and results of the function \( k(x) = \sqrt{x+2} \).