To determine the implied domain of the function, we need to analyze the constraints imposed by the mathematical expressions in the relation. The given relation is:

$\sqrt{1 - x} - 4y = 1$

Step-by-step process:

1. Square Root Function Constraint: The term $\sqrt{1 - x}$ exists only when $1 - x \geq 0$. This ensures that the expression under the square root is non-negative. Solving this inequality:

   $1 - x \geq 0 \implies x \leq 1$

   So, the first constraint is $x \leq 1$.

2. Solve for $y$: From the given equation $\sqrt{1 - x} - 4y = 1$, we can solve for $y$:

   $\sqrt{1 - x} - 4y = 1$ Add $4y$ to both sides: $\sqrt{1 - x} = 1 + 4y$ Solve for $y$: $y = \frac{\sqrt{1 - x} - 1}{4}$ There are no additional restrictions on $y$, since the square root constraint already limits $x$.

Implied Domain:

The only constraint comes from the square root, which requires $x \leq 1$. Additionally, the square root function has no lower bound on $x$, as any negative number under the square root would be undefined. So, the domain is:

$(-\infty, 1]$

Final Answer:

The implied domain of the function is $(-\infty, 1]$.

---

Would you like further clarification or more details on any part of the solution? Here are some related questions for deeper understanding:

1. What happens to the function's behavior as $x$ approaches 1?
2. Why can't the square root of a negative number exist in this context?
3. How would the domain change if we had a cube root instead of a square root?
4. What would happen if the expression involved both square and cube roots?
5. How do transformations like shifts or stretches affect the domain of square root functions?

Tip: Always