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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?

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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.

Find and sketch the domain of the function f(x, y) = √(y - 4x - 3)

Discover how to find and sketch the domain of the multivariable function f(x, y) = √(y - 4x - 3). The problem requires solving inequalities to determine valid values of y for given x, which forms a half-plane above the line y = 4x + 3. Ideal for students in Grades 9-12.

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