Questions: At what points is the following function continuous? f(x) = sqrt(2x + 3) The set of x-values where the function is continuous is . (Simplify your answer. Type your answer in interval notation.)

Solution Steps

The function given is \( f(x) = \sqrt{2x + 3} \). For the square root function to be defined, the expression inside the square root must be non-negative. Therefore, we need to solve the inequality:

\[ 2x + 3 \geq 0 \]

To solve the inequality \( 2x + 3 \geq 0 \), we perform the following steps:

1. Subtract 3 from both sides:

   \[ 2x \geq -3 \]

2. Divide both sides by 2:

   \[ x \geq -\frac{3}{2} \]

The solution to the inequality \( x \geq -\frac{3}{2} \) indicates that the function is defined for all \( x \) values greater than or equal to \(-\frac{3}{2}\). In interval notation, this is expressed as:

\[ \left[-\frac{3}{2}, \infty\right) \]

The function \( f(x) = \sqrt{2x + 3} \) is continuous wherever it is defined because the square root function is continuous for all non-negative arguments. Therefore, the function is continuous on the interval \(\left[-\frac{3}{2}, \infty\right)\).

Final Answer