To find the domain of the function $f(x) = \frac{3x+1}{4x+2}$, we need to determine where the denominator is not zero, since division by zero is undefined.

1. Set the denominator equal to zero and solve for $x$: $4x + 2 = 0$ $4x = -2$ $x = -\frac{1}{2}$ So, the function is undefined at $x = -\frac{1}{2}$.

2. Domain: The domain of $f(x)$ is all real numbers except $x = -\frac{1}{2}$. In interval notation, this is written as: $(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, +\infty)$

This means that the function is defined for all real numbers except $-\frac{1}{2}$.

Do you need more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How do we find the domain of more complex rational functions?
2. What happens graphically at the point where the denominator is zero?
3. How would the domain change if the denominator was a quadratic expression?
4. Can a rational function have any other types of domain restrictions besides division by zero?
5. What is the range of this particular function?

Tip: Always check the denominator of a rational function first when determining the domain.