Problem 8 - Basic Technical Mathematics with Calculus

Question

Find the domain and range of the given functions. $$F(r)=\sqrt{r+4}$$

Step-by-Step Solution

Answer

The domain is \([-4, \infty)\) and the range is \([0, \infty)\).

Step 1: Understanding the Function

The function given is a square root function: \( F(r) = \sqrt{r+4} \). To find the domain and range systematically, we need to consider the properties of the square root function, especially focusing on the radicand, which is the expression inside the square root, \( r+4 \).

Step 2: Finding the Domain

For \( F(r) = \sqrt{r+4} \), the quantity under the square root, \( r+4 \), must be greater than or equal to zero, since the square root of a negative number is not defined in the real number system. Solve the inequality: \( r+4 \geq 0 \). This simplifies to \( r \geq -4 \). Thus, the domain of \( F \) is all real numbers \( r \) such that \( r \geq -4 \). In interval notation, this is \([-4, \infty)\).

Step 3: Finding the Range

The value of \( F(r) \) is determined by the square of any non-negative number. Since \( F(r) = \sqrt{r+4} \), the smallest value of \( F(r) \) occurs when \( r = -4 \). Substitute \( r = -4 \) into the function: \( F(-4) = \sqrt{-4+4} = \sqrt{0} = 0 \). As \( r \) becomes larger, \( r+4 \) becomes larger, meaning \( F(r) \) also increases. Thus, the range of \( F \) is \([0, \infty)\).

Key Concepts

Square Root Function

In mathematics, a square root function is a type of function that contains a square root of a variable expression. Specifically recognizing the form, it often looks like \( F(x) = \sqrt{x} \). However, in the provided function, \( F(r) = \sqrt{r+4} \), notice that the expression inside the square root, or the radicand, isn't just \( r \) but \( r+4 \).

Square root functions have certain characteristics that are important:

• The domain typically involves values that ensure the radicand is non-negative, as square roots of negative numbers are not defined in the real number system.
• They're always non-negative outputs because the principal square root (the positive one) is used, resulting in \( F(r) \geq 0 \).
Understanding these basic properties helps in analyzing and solving problems related to square root functions.

Inequalities

To determine the conditions under which a square root function is defined, understanding inequalities is crucial. An inequality indicates a relationship that one expression is not exactly equal to another and may be either greater or less.

For the function \( F(r) = \sqrt{r+4} \), the expression inside the square root should satisfy:

• \( r+4 \geq 0 \), which implies that \( r \geq -4 \).

Remember, solving inequalities often means "finding out which values can replace the variable" to make the inequality true. This solution ensures that \( r \) is contained within the domain of the function.

Interval Notation

Interval notation is a mathematical notation used to describe a set of numbers along a number line. It provides a concise way to represent the domain or range of a function.

For the function \( F(r) = \sqrt{r+4} \), the domain expressed in interval notation based on the inequality \( r \geq -4 \) is \([-4, \infty)\). Here:

• The bracket "[" indicates that -4 is included in the domain (because \( r \geq -4 \)).
• The parenthesis ")" indicates infinity is unbounded, meaning there is no upper limit.

Similarly, for the range, since the smallest value of \( F(r) \) is 0, as \( r+4 \) is non-negative, the range starts from 0 to infinity, expressed as \([0, \infty)\). Understanding this notation helps in defining precise limits for functions and their solutions.