Problem 19 - Calculus with Concepts in Calculus

Question

Find the domain of the function. $$ f(x)=\sqrt{x+2} $$

Step-by-Step Solution

Answer

The domain of the function is \([-2, \infty)\).

Step 1: Understanding the function

The given function is \( f(x) = \sqrt{x + 2} \). This is a square root function.

Step 2: Identify constraints of the square root

For the square root function \( \sqrt{x + 2} \) to be defined over real numbers, the expression under the square root must be non-negative. This is because the square root of a negative number is not a real number.

Step 3: Set up inequality

To find when the expression \( x + 2 \) is non-negative, set up the inequality: \( x + 2 \geq 0 \).

Step 4: Solve the inequality

Solve the inequality \( x + 2 \geq 0 \) by subtracting 2 from both sides: \( x \geq -2 \).

Step 5: Write the domain

The domain of the function \( f(x) = \sqrt{x + 2} \) consists of all values of \( x \) that satisfy the inequality \( x \geq -2 \). Thus, the domain is \( [-2, \infty) \).

Key Concepts

Square Root Function

The square root function is an essential concept in mathematics, often encountered in various types of problems. A square root function is generally expressed as \( f(x) = \sqrt{x} \), where \( x \) is any real number for which the function is defined. The domain of a square root function is critical because it determines the set of input values for which the square root is a real number.

• The expression inside the square root, often called the 'radicand', must be non-negative (zero or positive) to ensure that the function outputs a real number.
• An important aspect of such functions is that they transform inputs into non-negative outputs, as the square root of a number is never negative in the realm of real numbers.
• For example, in the function \( f(x) = \sqrt{x+2} \), the radicand is \( x+2 \), which must be non-negative to ensure that the output is a real number.

By understanding these basic principles of square roots, you can effectively determine the domain of functions involving square roots.

Inequalities

Inequalities are a foundational tool to analyze and describe sets of numbers that satisfy certain conditions. They often appear in mathematics to define domains, especially in the context of square root functions.

• The inequality \( x + 2 \geq 0 \) means that \( x + 2 \) should be zero or more. This ensures the expression inside a square root is non-negative.
• Inequalities can be solved just like equations by performing operations that maintain the inequality's truth, such as adding or subtracting the same number from both sides.
• For the inequality \( x + 2 \geq 0 \), solving it involves simple arithmetic: subtract 2 from both sides to find that \( x \geq -2 \). This solution tells you that \( x \) must be greater than or equal to \(-2\).

Mastering these basic moves with inequalities is key to analyzing more complex mathematical expressions and determining domains.

Real Numbers

Real numbers encompass a vast array of numbers, including integers, fractions, and irrationals. They form the real number line, representing all possible magnitudes without imaginary components.

• Critical to understanding function domains is knowing that all values in these intervals—such as \([-2, \infty)\) in problems like our square root function—are real numbers.
• Real numbers are essential because all computations and problems typically deal with this set unless specified otherwise, like in complex number problems.
• In the context of square roots, imagining numbers is outside real numbers unless you encounter negative radicands.

Understanding real numbers helps students grasp why certain expressions limit function domains and ensures solutions are rooted in real-world applications.