Problem 10 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Add 2 to \( x \) and then find the square root.

Step 1: Identify the Function

The given function is \( k(x) = \sqrt{x+2} \). This function involves a square root and an expression inside the square root.

Step 2: Identify the Inner Expression

Within the square root, we have the expression \( x + 2 \). This is what we need to consider to understand its effect.

Step 3: Understand the Operation

The function applies a square root to the expression \( x + 2 \). This means we are asked to find the number which, when squared, equals \( x + 2 \).

Step 4: Express in Words

The function \( k(x) = \sqrt{x+2} \) can be described in words as "Take the input \( x \), add 2 to it, and then find the square root of the result."

Key Concepts

Function Notation

Function notation is a way to easily represent a mathematical relationship between variables. It uses symbols and letters to show how one quantity depends on another. In function notation, the general format is \( f(x) \), where \( f \) is the name of the function and \( x \) is the input variable. The rules within the function help us determine the output based on given inputs.

In the example of \( k(x) = \sqrt{x+2} \), \( k \) is the function's name, and \( x \) represents any value we choose to input into the function. The notation is a convenient way to express complex mathematical operations easily. It allows for the clarity of communication when discussing or calculating the behavior of functions.

Key aspects of function notation include:

• Function Name (\( f, g, h, k \), etc.): Represents the type of operation or rule applied.
• Input Variable (\( x \), for example): Denotes the value you provide to the function.
• Expression of Relationship: Shows how to obtain the output from the input.

Function notation thus simplifies the process of mapping inputs to outputs, making it easier to understand and manipulate mathematical expressions.

Mathematical Expressions

Mathematical expressions are combinations of symbols and numbers that represent a particular value or computation. They are like sentences in mathematics that convey operations to be performed with specified values. These expressions might include variables, constants, operators, and functions.

In the given function \( k(x) = \sqrt{x+2} \), the mathematical expression inside the square root, \( x + 2 \), is crucial. It indicates that for any value of \( x \), we add 2 before taking the square root of the result.

Most expressions include these elements:

• Variables: Symbols such as \( x \) that can take on multiple values.
• Operators: Signs like \(+, -, \times, \div, \sqrt{}\) which denote operations.
• Constants: Fixed numbers, in this case, the number 2.
• Functions: Such as \( \sqrt{} \), express specific operations like finding a square root.

Understanding mathematical expressions means knowing how these parts come together to describe relationships or operations that solve for specific outcomes. When translating a function into words, recognizing these components helps clarify the process.

Interpreting Functions

Interpreting functions involves understanding what a function does with its input values and determining the meaning behind the operations it performs. It requires analyzing both the notation and the expressions within the function to translate them into real-world contexts or verbal descriptions.

For the function \( k(x) = \sqrt{x+2} \), interpreting it means considering what the operations - addition and square root - imply for any input \( x \). You start by adding 2 to your chosen value of \( x \), which shifts your input by 2 units. Then, take the square root of this new value, finding a number that, when multiplied by itself, gives back \( x + 2 \).

Steps to interpret functions include:

• Identify the overall operation (e.g., square root).
• Determine the expression within the operation (e.g., \( x + 2 \)).
• Articulate the function in a verbal format to grasp its effect clearly.

By interpreting functions, you transform abstract mathematical symbols into comprehensible actions or processes, enabling better understanding and application of the function in different scenarios.