Questions: Find the domain of the function using interval notation f(x)=(3x+1)/(4x+2)

Transcript text: Find the domain of the function using interval notation \[ f(x)=\frac{3 x+1}{4 x+2} \]

Solution

Solution Steps

To find the domain of the function \( f(x) = \frac{3x + 1}{4x + 2} \), we need to determine the values of \( x \) for which the function is defined. The function is undefined when the denominator is zero. Therefore, we need to solve the equation \( 4x + 2 = 0 \) to find the values of \( x \) that make the denominator zero. The domain will be all real numbers except these values.

Step 1: Identify the Denominator

To find the domain of the function \( f(x) = \frac{3x + 1}{4x + 2} \), we first identify the denominator of the function, which is \( 4x + 2 \).

Step 2: Solve for Values that Make the Denominator Zero

The function is undefined when the denominator is zero. Therefore, we solve the equation: \[ 4x + 2 = 0 \] Solving for \( x \): \[ 4x = -2 \implies x = -\frac{1}{2} \]

Step 3: Determine the Domain

The domain of the function is all real numbers except \( x = -\frac{1}{2} \). In interval notation, this is expressed as: \[ (-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty) \]