91Ó°ÊÓ

Square root functions are a type of function involving a radical expression. Here, our function is given as \( y = \sqrt{2x + 3} \). The square root function is defined only for values where the radicand, which is the expression inside the square root, is non-negative. This is because in real numbers, you cannot take the square root of a negative number without venturing into complex numbers.
For our function, the radicand is \( 2x + 3 \). Hence, to find where the function is continuous, we need it to be non-negative.
This means the expression \( 2x + 3 \) must be greater than or equal to zero. Solving this will determine the domain of the function, which also points to its continuity.

Inequalities are mathematical expressions involving the symbols \(<, >, \leq, \geq\). They are essential in determining the range of values for which the expression holds true. In our problem, the inequality \( 2x + 3 \geq 0 \) defines where the function \( y = \sqrt{2x + 3} \) is valid.
By solving this inequality -

• First, subtract 3 from both sides to get \( 2x \geq -3 \).
  
• Then, divide by 2 to isolate \( x \), resulting in \( x \geq -\frac{3}{2} \).

This solution tells us that the function is continuous for all \( x \) values greater than or equal to \(-\frac{3}{2}\). This approach ensures that the function remains within the set of real numbers.

In calculus, continuity of a function at a certain point means that there are no breaks, jumps, or holes at that point for the function. For a function \( f(x) \) to be continuous at a point \( c \), it must satisfy three main conditions:

• \( f(c) \) must be defined.
• The limit of \( f(x) \) as \( x \) approaches \( c \) must exist.
• The limit of \( f(x) \) as \( x \) approaches \( c \) must be equal to \( f(c) \).

For the square root function \( y = \sqrt{2x + 3} \), we test continuity over the interval \([-\frac{3}{2}, \infty)\). At every point in this interval, the function satisfies all the above criteria. As such, the function has no interruptions or discontinuities within this range. Understanding these concepts helps students determine not just where functions are defined, but also where they are smooth and unbroken.