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Set-builder notation is a method of specifying a set by describing the properties that its members must satisfy. In simple terms, it tells us what kind of numbers or elements belong to a specific group.

Let's take the example from our exercise: The function is based on the square root of \( x + 69 \). The set-builder notation for the domain of this function is written as \( \{ x \in \mathbb{R} \,|\, x \geq -69 \} \). Here's what each part means:

• The curly brackets \( \{ \} \) indicate that we are describing a set.
• \( x \in \mathbb{R} \) means \( x \) is an element of the set of all real numbers, represented by \( \mathbb{R} \).
• The vertical bar \( | \) reads as "such that". It's a condition for the numbers in the set.
• Lastly, \( x \geq -69 \) describes the condition that the numbers in the set must meet.

In short, this notation describes the set of all real numbers \( x \) that are greater than or equal to \(-69\). This is how we use set-builder notation to express domains of functions in mathematics.

Interval notation is another efficient way to represent the domain of a function and it's a favorite for its simplicity. It provides a start and an end to indicate all numbers within that range, including or excluding specific endpoints.

For the function \( f(x)=\sqrt{x+69} \), we've established that \( x + 69 \) must be non-negative, leading to \( x \geq -69 \). In interval notation, we write this domain as \([-69, \infty)\). Here's how to read it:

• The square bracket \([\) on \(-69\) denotes that \(-69\) is included in the domain.
• The infinity symbol \(\infty\) accompanied by a parenthesis \()\) shows that the domain extends indefinitely in the positive direction, but \(\infty\) itself is not included because infinity is a concept rather than a specific number.

This way, interval notation compacts the description into an easy-to-read format, showing all numbers \( x \) starting from \(-69\) and stretching towards positive infinity.

Square root functions are fascinating because they deal with the principal square root or the non-negative root of a number. In mathematical terms, the square root function \( \sqrt{x} \) is defined only for non-negative values of \( x \).

When you see a function like \( f(x)=\sqrt{x+69} \), it implies a limitation. Specifically, the function takes only non-negative numbers for \( x+69 \). That's why \( x+69\geq0\), translating to \(x\geq-69\). This condition ensures that you're always working with defined, real-number outputs.

This property shapes such functions to generally appear only in the first quadrant when plotted on a graph, reflecting the characteristic that \( \sqrt{x} \) cannot assume negative input values for real numbers. Recognizing these aspects helps when calculating domains and understanding what inputs a function can accept.