The problem provided asks to determine the domain of the function:

$g(x) = \frac{\sqrt{5x - 3}}{2 - 4x}$

Step 1: Conditions for the Square Root

For the square root to be defined, the expression inside must be non-negative:

$5x - 3 \geq 0$

Solving this inequality:

$5x \geq 3 \implies x \geq \frac{3}{5}$

Step 2: Conditions for the Denominator

The denominator must not be zero to avoid division by zero:

$2 - 4x \neq 0$

Solving this equation:

$4x \neq 2 \implies x \neq \frac{1}{2}$

Step 3: Combine Conditions

The domain must satisfy both conditions:

1. $x \geq \frac{3}{5}$
2. $x \neq \frac{1}{2}$

Since $\frac{1}{2} < \frac{3}{5}$, this condition is already excluded. Thus, the domain is:

$x \geq \frac{3}{5}$

So the domain of the function is $x \in \left[ \frac{3}{5}, \infty \right)$.

Would you like further explanation or details on any step?

Related Questions:

1. What happens to the function if the square root is negative?
2. How does the denominator affect the domain in rational functions?
3. Can the function be continuous at $x = \frac{1}{2}$?
4. What is the domain of a general square root function?
5. How can you graphically interpret the domain restrictions?

Tip:

Always check for both the numerator and denominator constraints when dealing with rational functions.