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The domain of a function is the set of all possible input values (x-values) that the function can accept. In this case, our function has two parts: one for values where x is less than 0, and another for values where x is greater than or equal to 0. Both parts together cover all possible real numbers. Hence, the domain of this function is all real numbers, which we write as \((-\infty, \infty)\).

On the other hand, the range is the set of all possible output values (y-values) that the function can produce. For our function, when x is less than 0, the output (y-value) is always 3. When x is greater than or equal to 0, the output is 3 plus the square root of x. As x gets larger, so does the output. Hence, the range starts from 3 and goes to infinity. This can be written as \[ [3, \infty) \].

When it comes to graphing functions, it's important to plot ordered pairs on a coordinate plane. To do this, you need to carefully analyze the function given. Our function has two parts, and we'll graph each one separately before combining them.

For values of x less than 0, the function is a constant 3. This means that the graph will be a horizontal line at y=3 for all negative x-values. For x greater than or equal to 0, the function is 3 plus the square root of x. This part of the graph will start at the point (0, 3) and curve upwards as x increases. Points like (1, 4) and (4, 5) help visualize this curve.

To graph the whole function, plot all the ordered pairs: (-3, 3), (-2, 3), (-1, 3), (0, 3), (1, 4), and (4, 5). Connect the points for x less than 0 with a straight horizontal line, and connect the points for x greater than or equal to 0 with a smooth curve following the shape of y=3 + \( \sqrt{x} \). This method ensures a clear and accurate graph of the function.

Ordered pairs are simply pairs of numbers that represent points on a graph, written in the form (x, y). To fully represent a piecewise function graphically, we need to find these ordered pairs for different sections of the function.

In our case, we first looked at the section of the function where x is less than 0. Any x-value in this section produces a y-value of 3. We chose x-values like -3, -2, and -1 to form the ordered pairs (-3, 3), (-2, 3), and (-1, 3).

For the section where x is greater than or equal to 0, we used the function y = 3 + \( \sqrt{x} \). Choosing x-values like 0, 1, and 4, we calculated the corresponding y-values to form ordered pairs: (0, 3), (1, 4), and (4, 5).

Combining these ordered pairs helps us plot the function more easily and see how it behaves for all values of x, allowing for a clear understanding of the entire function's shape and range.