Problem 11 - Precalculus: Mathematics for Calculus

Question

Find the domain of the expression. $$\sqrt{x+3}$$

Step-by-Step Solution

Answer

Domain is \([-3, \infty)\).

Step 1: Understand the Problem

The expression is \( \sqrt{x+3} \). We are asked to find its domain. This means we need to determine for which values of \( x \) the expression is defined.

Step 2: Identify Restrictions for the Square Root Function

Recall that the square root function \( \sqrt{u} \) is only defined when \( u \geq 0 \). Hence, for \( \sqrt{x+3} \), the expression \( x+3 \) must be greater than or equal to 0.

Step 3: Solve the Inequality

Set up the inequality \( x + 3 \geq 0 \) and solve for \( x \). Subtract 3 from both sides to find \( x \geq -3 \).

Step 4: Write the Domain in Interval Notation

The solution from the inequality gives us the domain. In interval notation, the domain is \([-3, \infty)\), meaning \( x \) can be any real number greater than or equal to \(-3\).

Key Concepts

Square Root Function

The square root function is a common mathematical function we encounter often, denoted by the symbol \( \sqrt{} \). It represents a number that, when multiplied by itself, gives the original number underneath the square root symbol.​

The important thing about the square root function is that it only works when the number under the square root is zero or positive. This is because the square root of a negative number is not defined within the set of real numbers. Instead, it falls into the category of complex numbers, which is a topic for another day.

• For example, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).
• If we try \( \sqrt{-1} \), it does not have a real number solution.

The expression \( \sqrt{x+3} \) needs the part inside the square root, \( x+3 \), to be zero or positive. So, we begin our journey by finding these values.

Inequality Solving

Solving inequalities is like solving equations, but there are a few extra rules to remember. Inequalities show a relationship where one side is larger or smaller than the other.

Consider the inequality \( x + 3 \geq 0 \), which arose from ensuring the term inside the square root is non-negative. Our task here is to rearrange this inequality and find the values of \( x \) that make it true.

Here's how we proceed:

• You have \( x + 3 \geq 0 \), suggesting that the smallest \( x \) can be is one that makes \( x + 3 \) exactly zero.
• Subtract 3 from both sides: \( x \geq -3 \).

This simple operation reveals that any number \( x \) that is greater than or equal to \(-3\) will keep the expression \( x+3 \) greater than or equal to zero, thereby making the square root expression valid.

Interval Notation

Interval notation is a shorthand way to write the solution set of an inequality. It tells us the range of values that a variable like \( x \) can take.

In our case, the inequality \( x \geq -3 \) translates to interval notation as \([-3, \infty)\).

• The square bracket \([ \) indicates that \(-3\) is included in the interval, meaning \( x \) can be exactly \(-3\).
• The parenthesis \(( \)) next to \( \infty \) shows that infinity is not a number we reach, but a concept indicating the numbers keep increasing.

Using interval notation provides a clear and concise way to express the entire set of values that qualify for our solution. It ensures anyone reading your results immediately understands the boundaries of the domain for your function.