Find the domains of the functions.
`(1)/(sqrt(2x+3))`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{2x + 3}} \), we need to determine the values of \( x \) for which the function is defined.

### Step 1: Identify the conditions for the function to be defined
The function \( f(x) \) involves a square root in the denominator. For the function to be defined, the expression inside the square root must be positive, as the square root of a negative number is not defined in the set of real numbers. Therefore, we need:
\[ 2x + 3 > 0 \]

### Step 2: Solve the inequality
To solve the inequality \( 2x + 3 > 0 \), we can isolate \( x \):
\[ 2x > -3 \]
Now, divide both sides by 2:
\[ x > -\frac{3}{2} \]

### Step 3: Write the domain in interval notation
The solution \( x > -\frac{3}{2} \) indicates that \( x \) can take any value greater than \(-\frac{3}{2}\). In interval notation, this is expressed as:
\[ \text{Domain} = \left(-\frac{3}{2}, \infty\right) \]

### Final Answer
Thus, the domain of the function \( f(x) = \frac{1}{\sqrt{2x + 3}} \) is:
\[ \text{Domain} = \left(-\frac{3}{2}, \infty\right) \]

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