Problem 20 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

No, the equation \(x + y^{2} = 4\) does not represent y as a function of x.

Step 1: Express y in terms of x

For an equation to be a function of y in terms of x, it should be possible to write it in the form \(y = f(x)\). We will therefore try to express y in terms of x. Isolate \(y^{2}\) on one side of the equation by subtracting x from both sides, which gives us \(y^{2} = 4 - x\).

Step 2: Solve for y

The equation can now be solved for y. The square root of \(y^{2}\) will be \(y\), but we must remember that the square root can have both positive and negative values. Therefore, we get \(y = \sqrt{4 - x}\) and \(y = -\sqrt{4 - x}\). This means for some values of x, there could be two corresponding values of y.

Step 3: Conclusion

Since for some x values there are two possible y values, the equation \(x + y^{2} = 4\) does not represent y as a function of x as per the definition of a function, which states that for each x-value, there should be exactly one y-value.

Key Concepts

Relation vs. Function

When talking about mathematics, it's essential to understand the difference between a relation and a function. A relation is simply any set of ordered pairs. For instance, you can have a relation without any restrictions regarding the values of each pair.

A function is a particular type of relation that follows stricter rules. Specifically, each input value \(x\) must pair with exactly one output value \(y\). This means that if you graph a function, any vertical line should intersect the graph at most once. This is known as the "vertical line test."

To determine if an equation represents \(y\) as a function of \(x\), check if every \(x\) leads to only one \(y\). If any \(x\) yields more than one \(y\), then the relation is not a function. That's the core difference between a relation and a function.

Solving for y

The key to identifying if an equation might be a function is often being able to express \(y\) in terms of \(x\). For the equation \(x + y^2 = 4\), begin by isolating the \(y\)-related term.

Subtract \(x\) from both sides to get \(y^2 = 4 - x\). Now, solve for \(y\) by taking the square root of both sides. Here, you encounter critical insight: square roots typically give two solutions—positive and negative.

Therefore, you end up with two equations: \(y = \sqrt{4 - x}\) and \(y = -\sqrt{4 - x}\). This demonstrates that for each valid \(x\), there are two possible \(y\) values, further suggesting it does not fit the definition of \(y\) being a function of \(x\). Solving for \(y\) has opened the path to understanding this concept.

Determining Functions

Once \(y\) has been solved in terms of \(x\), determining whether you have a function involves checking whether a consistent \(x\) consistently delivers one \(y\). Our latest example turned \(x + y^2 = 4\) into two potential \(y\)-values for each \(x\). This violates the fundamental rule that a function can only associate each \(x\) with a single \(y\).

To identify functions successfully, you'll frequently use graphical or algebraic methods:

• Perform a vertical line test on the graph of the equation. If any vertical line meets the graph more than once, it is not a function.
• Analyze the form of the equation. If solving for \(y\) gives more than one possible expression, as it does in this case, the relation may not be a function.

Recognizing functions involves both logic and foundational math skills, ensuring you grasp the heart of how inputs relate to outputs.