To find the domain of the function $f(x, y) = \sqrt{y - 4x - 3}$, we need to ensure that the expression inside the square root is non-negative. This is because the square root function is only defined for values greater than or equal to zero (in the set of real numbers).

Step 1: Set the expression inside the square root to be non-negative.

$y - 4x - 3 \geq 0$

Step 2: Solve for $y$.

$y \geq 4x + 3$

Step 3: Interpret the result.

The domain of the function is the set of all points $(x, y)$ where $y \geq 4x + 3$. This represents a half-plane in the coordinate plane, including the boundary line $y = 4x + 3$.

Step 4: Sketch the domain.

1. The line $y = 4x + 3$ has a slope of 4 and a y-intercept of 3.
2. The domain consists of all points on or above this line.

To sketch:

• Draw the line $y = 4x + 3$.
• Shade the region above the line to represent the valid values of $y$.

Would you like me to generate a visual sketch of the domain?

Do you have any questions or would you like more details on any part? Here are some related questions to explore:

1. What is the significance of the boundary line $y = 4x + 3$ in the domain?
2. How would the domain change if the function had $f(x, y) = \sqrt{4x + 3 - y}$?
3. What happens to the domain if the square root function is replaced with a cube root?
4. How do we determine the range of the function $f(x, y)$?
5. How do constraints like $y \geq 4x + 3$ affect multi-variable functions in general?

Tip: The domain of a function involving a square root is determined by ensuring the argument of the square root is non-negative.