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Understanding intervals of continuity helps us figure out where a function behaves nicely without any breaks. For a function to be continuous on a certain interval, we must be able to draw it on this interval without lifting our pencil. Continuity requires that there are no jumps, holes, or breaks in the graph.
In the exercise, the function's continuity is determined by where it is defined. The expression under a square root needs to be non-negative for the function to be defined.
Thus, after solving the inequality, we find that the function \(f(x) = x\sqrt{x+3}\) is continuous on

• \([-3, \infty)\)

This interval means that the function remains continuous and defined for all values from -3 to infinity, including -3 itself.

Square root functions are those that involve the square root of an expression, like \(\sqrt{x+3}\). These functions have unique properties, particularly in how they handle domain restrictions due to the square root's nature.
For any square root function \(\sqrt{x}\), the expression inside must be greater than or equal to zero. This condition ensures that the result is a real number. For the function \(x\sqrt{x+3}\), it means solving \(x+3 \geq 0\) to find the domain.
This gives us \(x \geq -3\), ensuring that when \(x\) is plugged in, the square root provides a real and defined output. Avoiding negative results under the square root is key to preventing undefined areas in the function.

Inequalities are vital to determine where a function is continuous or defined. They help us identify critical boundaries where the function's behavior changes. In this problem, we solved the inequality \(x+3 \geq 0\) to find the domain of the function.
Here's how we do it:

• Subtract 3 from both sides to get \(x \geq -3\).

What's essential about inequalities:

• They show where expressions are valid for real-world values.
• Provide the domain and points of continuity for functions.

By solving the inequality, we identified that our function remains valid and continuous from -3 to infinity. This step is crucial, as it sets the stage for understanding and working with functions including finding limits and analyzing behavior.