Problem 22 - Precalculus with Limits

Question

In Exercises 19-36, determine whether the equation represents \(y\) as a function of \(x\). \(y - 4x^2 = 36\)

Step-by-Step Solution

Answer

Yes, the equation \(y = 4x^2 + 36\) represents \(y\) as a function of \(x\).

Step 1: Rearrange the Equation

The equation is given as \(y - 4x^2 = 36\). The first step is to rearrange it in the form \(y =f(x)\). Add \(4x^2\) to both sides of the equation to isolate \(y\). Hence, the equation becomes: \(y = 4x^2 + 36\)

Step 2: Verify the Equation

Now it's important to verify if the equation represents \(y\) as a function of \(x\). Here, \(y\) is being represented as the sum of \(4x^2\) and 36. In this equation, for every value of \(x\), there is a unique value of \(y\).

Step 3: Conclusion: Function Verification

Therefore, the equation \(y = 4x^2 + 36\) does represent \(y\) as a function of \(x\). This is because every \(x\) value maps to one and only one \(y\) value, fulfilling the definition of a function.

Key Concepts

Equation Manipulation

Equation manipulation involves rearranging an equation to a simpler or more useful form. In the given exercise, we started with the equation \(y - 4x^2 = 36\). To determine if it represents \(y\) as a function of \(x\), we need to isolate \(y\) on one side of the equation. This means that we have to apply simple arithmetic operations to both sides.

• We added \(4x^2\) on both sides to cancel out the \(-4x^2\) on the left side.
• This operation helps in isolating \(y\) which gives us \(y = 4x^2 + 36\).

Thus, the equation is now in the "function form" \(y = f(x)\), making it easier to analyze and verify if it represents a function.

Function Verification

Function verification is a process to determine if an equation fulfills the definition of a function. For the equation we manipulated, \(y = 4x^2 + 36\), we need to check if, for every possible value of \(x\), there is exactly one corresponding value of \(y\).In this case, let's break down the expression:

• \(4x^2\) is a parabola opening upwards, implying it is well-behaved and continuously mapped for every \(x\).
• Adding 36 simply shifts the parabola up by 36 units.

For any real number \(x\), there is a unique output \(y\), which confirms that this equation indeed represents a function.

Unique Value Mapping

Unique value mapping is a critical feature of functions. In the context of the equation \(y = 4x^2 + 36\), it is important to understand how each input \(x\) relates to the output \(y\).A function by definition must map each \(x\) to exactly one \(y\). Here's why our equation fits:

• For every \(x\), \(4x^2\) calculates one specific number. Multiplying \(x\) squared by 4 ensures that no two different values of \(x\) can produce the same value of \(4x^2\).
• Adding 36 subsequently guarantees that the output \(y\) is simply the result of the above calculation + 36, which means each \(x\) results in a unique \(y\).

This uniqueness is what makes \(y = 4x^2 + 36\) a function, fulfilling the criteria that each input corresponds to precisely one output.