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When discussing functions, understanding the domain is crucial. The domain of a function refers to all the possible input values (usually x-values) that the function can accept without resulting in any undefined or illegal operations. For example, in the function given by \( f(x) = \sqrt{x-2} \), the expression under the square root symbol must be non-negative. This is because you cannot take the square root of a negative number within the set of real numbers. Therefore, we set the inequality:

• \( x - 2 \geq 0 \)
• Solving this inequality gives \( x \geq 2 \)

This tells us that the domain of the function \( f(x) \) includes all real numbers starting from 2 and going to infinity. By identifying the domain, we know that any value of \( x \) less than 2 will not satisfy the function and should not be used in plotting.

The square root function is an important mathematical concept. Typically written as \( f(x) = \sqrt{x} \), this function represents the principal (or positive) square root of the input x. One of its key characteristics is that it gradually becomes less steep as x increases.Consider the transformation in our exercise, \( f(x) = \sqrt{x-2} \). Here, the graph of the square root function is shifted horizontally. The "-2" inside the square root function indicates a shift to the right by 2 units on the x-axis. Thus, the graph doesn't start at zero, but at the point where \( x = 2 \). From that point forward, the function smoothly increases.This behavior makes the square root function useful in many real-world problems where negative outputs don't make sense. It inherently starts at zero and only gives positive results, modeling scenarios where quantities can't be negative.

Plotting points is a foundational skill in graphing functions. The process begins with selecting values from the function's domain. For \( f(x) = \sqrt{x-2} \), we choose x-values such as 2, 3, 4, and 5. These values are within our domain, which is \( x \geq 2 \). We then calculate the corresponding y-values using the function:

• At \( x = 2 \), \( y = \sqrt{2-2} = 0 \)
• At \( x = 3 \), \( y = \sqrt{3-2} = 1 \)
• At \( x = 4 \), \( y = \sqrt{4-2} = \sqrt{2} \)
• At \( x = 5 \), \( y = \sqrt{5-2} = \sqrt{3} \)

Next, plot these points on the graph. Place each point where its continuous visuals meet on the x-axis and y-axis. It is essential to plot these points accurately as they form the backbone of the function's graph. After plotting, draw a smooth curve that connects all points to complete the function's sketch.

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves that represent functions. It is composed of two axes that intersect at a right angle. These two axes divide the plane into four quadrants:

• The horizontal axis is called the x-axis
• The vertical axis is called the y-axis

In graphing functions, the coordinate plane serves as the stage upon which the drama of mathematics unfolds. Each point on the plane is described by a pair of numbers \((x, y)\). The x-value describes the position along the horizontal axis, and the y-value describes the position along the vertical axis.Using a coordinate system allows us to visually interpret and understand the behavior of functions. When we sketched our function \( f(x) = \sqrt{x-2} \), it became apparent how the function behaves and how it transitions smoothly across the domain. This visual representation provided by the coordinate plane makes complex functions easier to comprehend.