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When we talk about the "domain of a function," we're asking about all the possible input values (specifically, the x-values) that we can put into a function without running into any issues, like division by zero or taking the square root of a negative number.
In the case of the inequality \( y \geq \sqrt{5x - 8} \), to find the domain, we need to look at the expression inside the square root: \( 5x - 8 \). This expression needs to be non-negative for the square root to exist.
This means we need to solve the inequality:

• \( 5x - 8 \geq 0 \)
• \( 5x \geq 8 \)
• \( x \geq \frac{8}{5} \)

So in simple terms, the domain here is all values of \( x \) that are greater than or equal to \( \frac{8}{5} \). Any smaller \( x \)-values would make the expression under the square root negative, which isn't allowed. Understanding the domain is crucial as it tells us the region on the x-axis where our function is defined and can be graphed.

The square root function is a type of radical function that gives us the positive square root of a number or expression. It is symbolized by \( \sqrt{} \). When dealing with square root functions, it's important to remember that they only produce non-negative outputs.
For the inequality \( y \geq \sqrt{5x - 8} \), we're dealing with the square root of an expression, \( \sqrt{5x - 8} \). This function visually represents a curve that starts from the domain restriction \( x = \frac{8}{5} \) onwards.
Key characteristics include:

• It starts at the point where the expression inside the square root equals zero. For this example, it's at \( x = \frac{8}{5} \), \( y = 0 \).
• As \( x \) increases beyond \( \frac{8}{5} \), \( y \) also increases.
• It's always important to ensure the expression inside the radical is non-negative to be valid.

By graphing \( y = \sqrt{5x - 8} \), we define the boundary where the values of \( y \) are equivalent to the square root expression. Beyond this curve, the inequality indicates that \( y \) can also be greater.

Shading regions in graphs is a graphical way to represent solutions of inequalities. It helps visually identify where the inequality holds true on the graph. For this exercise with \( y \geq \sqrt{5x - 8} \), shading indicates all the points where \( y \) values are equal to or above the curve.
The steps to shade the correct region involve:

• Graphing the boundary: Here \( y = \sqrt{5x - 8} \) is the boundary line.
• Determining if a line should be solid or dashed: We use a solid line because the inequality includes equality (\( \geq \)). A dashed line is used if the inequality was strictly "greater than" (\( > \)).
• Choosing a test point: This helps decide which side of the line to shade. In this case, after testing a point like \( (2,0) \), if a point does not satisfy the inequality, we shade the opposite region to this point.

For \( y \geq \sqrt{5x - 8} \), after plotting and testing, we shade above the boundary line, which means all areas where \( y \) is greater than or equal to \( \sqrt{5x - 8} \). This shading visually solves the inequality, illustrating the region that contains all possible solutions.