Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Describe the interval(s) on which the function is continuous. \(f(x)=x \sqrt{x+3}\)

Step-by-Step Solution

Answer

The function is continuous on the interval \([-3, \infty)\].

Step 1: Identify where the function cannot be defined

The function \(f(x) = x\sqrt{x+3}\) will not be defined wherever the x-values make the expression under the square root negative. Here, this happens when \(x+3 < 0\). By solving this inequality, we find \(x < -3\). So, the function is not defined when \(x < -3\).

Step 2: Identify where the function is defined

When \(x+3 \geq 0\), the function is defined. So, solve the inequality for \(x\): \(x+3 \geq 0 \Rightarrow x \geq -3\). Therefore, the function is defined when \(x \geq -3\).

Step 3: State the interval of continuity

Based on the results from step 1 and step 2, it is clear that the function is continuous on the interval \([-3, \infty)\]. This includes -3 and all numbers greater than -3. Thus, the function is continuous wherever it is defined.

Key Concepts

Intervals of Continuity

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

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.

Solving Inequalities

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.