Problem 83 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$h(n)=\sqrt{n+2}$$

Step-by-Step Solution

Answer

The domain of the function \(h(n)=\sqrt{n+2}\) is \([-2, \infty)\).

Step 1: Identify the inequality to be solved

Since the expression inside the square root must be non-negative, we need to solve the inequality: $$ n + 2 \geq 0 $$

Step 2: Solve the inequality

To find the value of n, we'll isolate it by subtracting 2 from both sides of the inequality: $$ n \geq -2 $$

Step 3: Write the domain of the function in interval notation

The inequality tells us that n must be greater than or equal to -2. In interval notation, this is written as: $$ [-2, \infty) $$ So, the domain of the function \(h(n)=\sqrt{n+2}\) is all real numbers greater than or equal to -2.

Key Concepts

Understanding the Square Root Function

The square root function \( h(n) = \sqrt{n+2} \) is a type of function involving a radical, specifically a square root.
This function takes an input value \( n \) and applies the square root to the expression \( n + 2 \). A key characteristic of square root functions is that they only produce non-negative outputs when dealing with real numbers.

• For the square root to be defined, the expression inside must be zero or positive.
• So, \( n+2 \geq 0 \) ensures the square root has a real result.

This is why solving inequalities associated with square root functions is crucial for determining their domain.

Solving Inequality for Domain Restrictions

When working with square root functions, it's essential to solve inequalities to find the domain.
For \( h(n) = \sqrt{n+2} \), you need to ensure that \( n+2 \geq 0 \). Solving this inequality:

• Isolate \( n \) by subtracting 2 from both sides: \( n \geq -2 \).
• This tells us that any \( n \) greater than or equal to \(-2\) will make \( n+2 \) non-negative.

By solving this inequality, you ensure the function's output remains real and defined, which is crucial for further mathematical applications.

Expressing Domains Using Interval Notation

Interval notation is a concise way to express the set of numbers or domain that satisfy a condition.
For our function \( h(n) = \sqrt{n+2} \), the inequality \( n \geq -2 \) translates into interval notation. Here’s how it works:

• The solution \( n \geq -2 \) represents all numbers starting from \(-2\) and going to infinity.
• Written in interval notation as \([-2, \infty)\), where:
• Bracket \([\,\) includes \(-2\).
• Parenthesis \()\) signifies infinity is not a specific number to reach.

Interval notation provides a clear and efficient way to express the range of values for a function, making it widely used in mathematics.

Understanding Real Numbers Inequality

Real numbers inequality involves finding the limits where a function outputs real numbers.
In \( h(n) = \sqrt{n+2} \), ensuring the expression inside the root is non-negative, establishes the inequality \( n \geq -2 \). Here's why this matters:

• Real numbers are any numbers that aren't imaginary. They include integers, fractions, and decimals.
• An inequality solution like \( n \geq -2 \) indicates the boundary where the function's value is a real number.

By understanding this, you can predict the function's behavior and ensure it operates within the real number system, avoiding undefined or imaginary results.