Question

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

Short answer

The function \( f(x)=x \sqrt{x+3} \) is continuous on the interval \([-3, +\infty)\]

Step-by-step solution

Identify the Domain

The first step is to identify the domain, i.e., the set of all possible values of \( x \) for which the function is defined. Any real number can be plugged into \( x \) but \( x+3 \) must be greater than or equal to zero because you can’t take the square root of a negative number. Therefore:\[ x+3 \geq 0 \]Solving for \( x \) yields:\[ x \geq -3 \]This implies that the function is defined for all \( x \) in the interval \([-3, +\infty)\]

Determine Continuity

Since the given function is a product of two functions, \( x \) and \( \sqrt{x+3} \), which are both continuous on their respective domains, the product is also continuous on their common domain. So, the function \( f(x)=x \sqrt{x+3} \) is continuous wherever it is defined, i.e., on the interval \([-3, +\infty)\]

Present the Conclusion

Having identified the domain and based on the properties of continuity, it is concluded that the function \( f(x)=x \sqrt{x+3} \) is continuous on the interval \([-3, +\infty)\]