To find the domain of the function

$f(x) = \frac{5x - 6}{(2x - 3)(4x - 1)}$

we need to determine the values of $x$ that make the function undefined. This will occur when the denominator is zero.

1. Set the denominator equal to zero: $(2x - 3)(4x - 1) = 0$

2. Solve each factor for $x$:

   • For $2x - 3 = 0$: $x = \frac{3}{2}$
   • For $4x - 1 = 0$: $x = \frac{1}{4}$

The function is undefined at $x = \frac{3}{2}$ and $x = \frac{1}{4}$.

3. Exclude these values from the domain. In interval notation, the domain is: $(-\infty, \frac{1}{4}) \cup \left( \frac{1}{4}, \frac{3}{2} \right) \cup \left( \frac{3}{2}, \infty \right)$

Final Answer:

The domain of $f(x)$ in interval notation is: $(-\infty, \frac{1}{4}) \cup \left( \frac{1}{4}, \frac{3}{2} \right) \cup \left( \frac{3}{2}, \infty \right)$

Would you like further details, or have any questions?

Related Questions

1. How do you find the domain of a rational function in general?
2. What happens to the range of $f(x)$ if the domain has restrictions?
3. How do asymptotes relate to the values excluded from the domain?
4. How do you interpret vertical asymptotes on the graph of $f(x)$?
5. Can $f(x)$ have any other points where it is undefined besides where the denominator is zero?

Tip: To find the domain of a rational function, always focus on where the denominator equals zero, as these points make the function undefined.