Determining If Y² = X + 4 Is A Function

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Hey everyone! Let's dive into an interesting problem today that touches on the core concept of functions in mathematics. We're going to determine if the equation $y^2 = x + 4$ represents y as a function of x. This might sound a bit technical, but trust me, we'll break it down into easily digestible parts. To figure this out, the very first thing we need to do is isolate y. In other words, we need to rewrite the equation so it looks like y = something. This "something" will be an expression involving x. By getting y by itself, we can then analyze how its value changes as x changes. This is crucial for understanding whether the equation defines a function or not. So, let's grab our mathematical tools and get started on the process of solving for y!

Solving for y: The First Step

Okay, so we start with the equation $y^2 = x + 4$. Our goal, as we just discussed, is to get y all by itself on one side of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced! Here's where the concept of inverse operations comes into play. We have y squared, and the inverse operation of squaring is taking the square root. So, to undo the square, we'll take the square root of both sides of the equation. This gives us:

y2=±x+4\sqrt{y^2} = \pm \sqrt{x + 4}

Now, on the left side, the square root and the square cancel each other out, leaving us with just y. But here's a super important point: when we take the square root of both sides of an equation, we have to consider both the positive and negative roots. That's why we have the "±" (plus or minus) symbol in front of the square root on the right side. This is a crucial detail that will ultimately determine whether or not our equation represents a function. So, after taking the square root, our equation looks like this:

y=±x+4y = \pm \sqrt{x + 4}

This is where things get really interesting. We've successfully solved for y, but what does this "±" sign really mean? It means that for a single value of x, we might actually get two different values for y. Let's explore this further in the next section.

The Significance of ±: Two Values for y

Alright, let's really dig into what that "±" symbol is telling us. The equation $y = \pm \sqrt{x + 4}$ actually represents two separate equations:

  1. y=+x+4y = +\sqrt{x + 4}

  2. y=−x+4y = -\sqrt{x + 4}

This is super important! For any given x value (that makes the expression under the square root non-negative, of course), the first equation will give us a positive square root, and the second equation will give us a negative square root. Let's think about a specific example to make this crystal clear. Suppose we let x = 5. Plugging this into our equation, we get:

y=±5+4=±9=±3y = \pm \sqrt{5 + 4} = \pm \sqrt{9} = \pm 3

So, when x = 5, we get two possible values for y: y = 3 and y = -3. This is a key observation that will help us determine if we have a function or not. To drive this point home, consider this: if you were to plot this equation on a graph, you'd see that the vertical line at x = 5 intersects the graph at two points: (5, 3) and (5, -3). This visual representation reinforces the idea of two y values for a single x value. Now, let's connect this to the definition of a function.

The Vertical Line Test: A Quick Visual Check

Before we formally define a function, there's a handy visual tool we can use called the Vertical Line Test. This test provides a quick way to determine if a graph represents a function. The rule is simple: if any vertical line intersects the graph at more than one point, then the graph does not represent a function. Why does this work? Well, a vertical line represents a single x-value. If the line intersects the graph at more than one point, it means that for that x-value, there are multiple y-values. And as we'll see in the next section, this violates the definition of a function. Imagine drawing a vertical line through the graph of our equation, $y^2 = x + 4$. As we discussed earlier, for x = 5, we have two y-values (3 and -3). So, a vertical line at x = 5 would intersect the graph at two points. This tells us, even before we get to the formal definition, that this equation probably doesn't represent a function. But let's make sure by diving into the precise definition.

Definition of a Function: The Crucial Rule

Okay, let's get down to the nitty-gritty and define what a function actually is. In simple terms, a function is a relation (think of it as a set of ordered pairs (x, y)) where each input (x-value) is associated with exactly one output (y-value). This is the key rule! For every x you plug in, you should only get one y out. Think of a function like a vending machine. You put in a specific amount of money (the input, x), and you expect to get one specific snack or drink (the output, y). You wouldn't expect the machine to give you two different items for the same amount of money! Now, let's relate this back to our equation, $y^2 = x + 4$. We already saw that when we plug in x = 5, we get two different y values: 3 and -3. This means that our equation violates the definition of a function. We have one input (x = 5) leading to two different outputs (y = 3 and y = -3). This is a big no-no in the function world. So, the verdict is in: this equation does not represent y as a function of x.

Conclusion: Is $y^2 = x + 4$ a Function?

Let's wrap things up and make our final determination. We started with the equation $y^2 = x + 4$ and asked the question: does this equation represent y as a function of x? Through a step-by-step process, we solved the equation for y, revealing that $y = \pm \sqrt{x + 4}$. The presence of the "±" symbol was a big clue, indicating that for a single x value, we could potentially have two different y values. We then used the Vertical Line Test as a visual check, which further suggested that this equation wasn't a function. Finally, we revisited the formal definition of a function, which states that each input (x) must have exactly one output (y). Since our equation produced two y values for a single x value (like x = 5 giving us y = 3 and y = -3), we can confidently conclude that the equation $y^2 = x + 4$ does not represent y as a function of x. Understanding this concept is crucial for further mathematical studies, as functions are the building blocks of many advanced topics. So, well done for sticking with it and unpacking this problem with me!