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We are given the equation:

\[
64y^2 - x^2 = 64
\]

We are tasked with finding two explicit functions by solving for \( y \) in terms of \( x \). Let's proceed step-by-step:

### Step 1: Isolate \( y^2 \)
Move \( x^2 \) to the other side of the equation:

\[
64y^2 = 64 + x^2
\]

### Step 2: Divide by 64

\[
y^2 = \frac{64 + x^2}{64}
\]

This simplifies to:

\[
y^2 = 1 + \frac{x^2}{64}
\]

### Step 3: Take the square root

Now, take the square root of both sides to solve for \( y \):

\[
y = \pm \sqrt{1 + \frac{x^2}{64}}
\]

Thus, the two explicit functions are:

\[
y_1(x) = \sqrt{1 + \frac{x^2}{64}}
\]
and
\[
y_2(x) = -\sqrt{1 + \frac{x^2}{64}}
\]

These represent the two solutions for \( y \) in terms of \( x \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do these functions relate to the equation of a hyperbola?
2. What is the domain of the functions \( y_1(x) \) and \( y_2(x) \)?
3. Can this equation be expressed in a standard form of a conic section?
4. How do the graphs of these functions behave as \( x \) increases?
5. What does the geometry of the two functions tell us about the symmetry of the problem?

**Tip**: Always check if a function's domain or range is restricted by the original equation when solving algebraically.

Find two explicit functions by solving the equation 64y^2 - x^2 = 64 for y in terms of x.

Learn how to solve the equation 64y^2 - x^2 = 64 to find two explicit functions for y in terms of x. This problem covers key algebraic concepts such as isolating variables and applying square roots, suitable for students in Grades 10-12.

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