Question

Determine whether the equation defines y as a function of \(x .\) \(x^{2}-4 y^{2}=1\)

Short answer

No, the equation does not define y as a function of x.

Step-by-step solution

- Understand the Equation

We are given the equation: \[ x^2 - 4y^2 = 1 \]. We need to determine if this equation defines \(y\) as a function of \(x\).

- Isolate y^2

Rearrange the equation to isolate \(y^2\): \[ x^2 - 1 = 4y^2 \] Next, divide both sides by 4: \[ y^2 = \frac{x^2 - 1}{4} \]

- Solve for y

To solve for \(y\), take the square root of both sides: \[ y = \pm \sqrt{\frac{x^2 - 1}{4}} \] This means: \[ y = \frac{\sqrt{x^2 - 1}}{2} \] or \[ y = \frac{-\sqrt{x^2 - 1}}{2} \]

- Determine if it is a Function

For \(y\) to be a function of \(x\), each \(x\) value should correspond to exactly one \(y\) value. Since \( y = \pm \sqrt{\frac{x^2 - 1}{4}} \) provides two possible values for \(y\) (one positive and one negative) for the same \(x\) value, \(y\) is not a function of \(x\).

Key concepts

Isolate Variable

When solving algebraic equations, one key step can often be isolating the variable we are interested in.
In this problem, we want to see if the equation defines y as a function of x.
To begin, we start with the equation: \[ x^2 - 4y^2 = 1 \] Our goal is to isolate y.
We rearrange the equation to isolate the term containing y: \[ x^2 - 1 = 4y^2 \] This step was necessary to bring all the non- y terms to one side of the equation.
Afterwards, we divide by 4 to further isolate y^2: \[ y^2 = \frac{x^2-1}{4} \] Finally, we take the square root of both sides to solve for y: \[ y = \frac{eg egegeg egegegegeg \sqrt{x^2-1}}{2} \]
Important: When you isolate a variable, remember to perform the same operation on both sides of the equation.
This gets us closer to knowing if we have a function.

Solving Equations

Solving equations involves undoing operations to isolate the variable.
In our case, solving for y meant isolating y on one side of the equation. Let' s take a closer look: \[ x^2 - 4y^2 = 1 \ to \ x^2 - 1 = 4y^2 \]
We subtracted from both sides to move non-y terms left.
Next, divide by 4: \[ y^2 = \frac{x^2 - 1}{4} \]
Taking the square root simplifies the equation:
y is \[ \frac{eg egegeg\sqrt{x^2-1}}{2} \]

Remember, the process involves:

• Moving terms from one side to another, and
• Using inverse operations.
  Dividing and square roots separated y.

Function Criteria

A critical part is understanding the function criterion.
A function means each input (x) maps to one output (y).
In our example, \[y = \frac{- eg(x^2-1)}{2} \]
provides two outputs for a single x.