Let’s analyze the problem step-by-step.

Problem:

We are solving for $y$ in the equation:

$y^2 + x - 4 = 0$

Then we’ll determine which of the provided options correctly represents $y$.

Step 1: Rearrange the equation for $y^2$

$y^2 = 4 - x$

Step 2: Solve for $y$

Taking the square root of both sides:

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

Step 3: Match the answer with the provided options

The correct option is:

$\boxed{a. \ y = \pm \sqrt{4 - x}}$

Would you like a detailed explanation of the next steps, such as verifying that this satisfies the differential equation?

Related Questions:

1. What does it mean for a function to be an implicit solution to a differential equation?
2. How do we differentiate $y^2 + x - 4 = 0$ implicitly to find $\frac{dy}{dx}$?
3. Why is the solution restricted to $x \leq 4$ for real values of $y$?
4. Can $\pm \sqrt{4 - x}$ represent a single-valued function, or is it multi-valued?
5. How do we interpret the interval $(-\infty, 4)$ in this context?

Tip:

When solving implicit equations, always check the domain of the solution, as taking square roots imposes restrictions!