Finding the Domain of the Function: Simplifying 9/(3x+1/2)

To find the domain of the function $\frac{9}{3x+\frac{1}{2}}$, we need to determine for which values of $x$ the expression is defined.

1. The function is of the form $\frac{9}{3x+\frac{1}{2}}$. A fraction is undefined if its denominator is zero.

2. Set the denominator of the function equal to zero: $3x + \frac{1}{2} = 0$.

3. Solve the equation for $x$:

• First, subtract $\frac{1}{2}$ from both sides: $3x = -\frac{1}{2}$.
• Next, divide both sides by 3 to solve for $x$: $x = -\frac{1}{2} \times \frac{1}{3} = -\frac{1}{6}$.

4. The solution $x = -\frac{1}{6}$ indicates that at this value, the denominator becomes zero, making the function undefined. Therefore, $x \ne -\frac{1}{6}$.

Thus, the domain of the function is all real numbers except $x = -\frac{1}{6}$.

The correct answer is $x \ne -\frac{1}{6}$, which matches choice 2: $x\ne-\frac{1}{6}$.