Question

Use the Vertical Line Test to determine whether \(y\) is a function of \(x\). $$ y=\left\\{\begin{array}{ll} x+1, & x \leq 0 \\ -x+2, & x>0 \end{array}\right. $$

Short answer

Based on the Vertical Line Test, \(y\) is indeed a function of \(x\). This is because for every \(x\) value under consideration, there is exactly one corresponding \(y\) value, and there is no overlap between the ranges of \(x\) for different parts of the function.

Step-by-step solution

Identify the Different Pieces of the Function

First, identify the different parts of the piecewise function. Here, there are two parts: \(y=x+1\) for \(x \leq 0\) and \(y=-x+2\) for \(x>0\). Thus, the function will be evaluated differently on either side of \(x=0\).

Apply the Vertical Line Test to Each Piece

Now, use the Vertical Line Test on each piece of the function separately. Start with \(y=x+1\) for \(x \leq 0\). For any given \(x\) in this range, there is exactly one corresponding \(y\) value. Therefore, in this range, \(y\) is a function of \(x\). Then, turn to \(y=-x+2\) for \(x>0\). Again, for any given \(x\) in this range, there is exactly one corresponding \(y\) value. Therefore, in this range, \(y\) is also a function of \(x\).

Check for Overlapping Ranges

Since each part of the function passed the Vertical Line Test separately, check if there is overlap between the different parts. If there are values of \(x\) that are included in more than one part of the function definition, this could meaning the function doesn't pass the Vertical Line Test overall. However, the ranges \(x \leq 0\) and \(x>0\) do not overlap, so this is not a concern here.

Key concepts

Piecewise Function

Understanding the concept of a piecewise function is crucial for students diving into calculus education. A piecewise function is a mathematical expression defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Imagine it as a Chameleon, changing its behavior (in this case, the rule of calculation) based on the environment (the domain intervals).

In the given exercise, we encounter a typical example: \( y = \left\{ \begin{array}{ll} x+1, & x \leq 0 \ -x+2, & x>0 \end{array} \right. \) This piecewise function has two expressions, \( y=x+1 \) for \( x \leq 0 \) and \( y=-x+2 \) for \( x>0 \)—both defining how \( y \) is calculated in different intervals of \( x \). The output for a specific \( x \) is determined solely by the interval it lies in, providing a clear and split methodology for evaluation.

To supplement student understanding, it's instructive to visualize these functions. Plotting each sub-function on a graph within their respective intervals would highlight how a piecewise function can consist of distinct lines or curves that seamlessly connect at their borders. This visualization complements the analytical approach and can aid in better grasping the concept.

Function of x

When learning about functions in calculus education, the phrase 'function of \( x \)' frequently surfaces. But what does it entail? Essentially, a function of \( x \) describes a relationship where every input \( x \) from the domain has exactly one output. It's like a strict rule in a boarding school: for each student (\( x \)), there's a unique bunk (\( y \)) assigned to them.

In the Vertical Line Test used in our exercise, we scrutinize the graph of a function to confirm if each \( x \) value has a single corresponding \( y \) value. If any vertical line intersects the function's graph at more than one point, then that graph does not represent a function of \( x \). This test is a quick visual inspection tool.

The sub-functions in our piecewise function passed this test within their defined intervals. The first part, \( y=x+1 \), has no confusion—for each \( x \) at or below zero, there’s one output \( y \). Likewise, for the second part, \( y=-x+2 \), every \( x \) greater than zero maps to a unique \( y \). Together, they play well to describe a function of \( x \), since there is no ambiguity in outputs for any \( x \) in the domain.

Calculus Education

The realm of calculus education encompasses a variety of essential concepts and techniques used to understand and apply mathematical principles. One primary goal in this field is to break down complex ideas, like piecewise functions and the Vertical Line Test, into digestible parts. Much like a complex machine is understood better by studying its individual components, calculus becomes more approachable when deconstructed into its fundamental pieces.

Exercises like the one explored are not just about getting the right answer. They're about nurturing a deep understanding of the relationship between a function and its domain, the difference between continuous and discontinuous graphs, and the significance of the Vertical Line Test as a tool to identify functions graphically. These exercises train students to think critically and to visualize mathematical concepts, which are invaluable skills both inside and outside academia.

By emphasizing the practical applications of these concepts, such as engineering or economics, calculus education can motivate students to see the relevance and importance of understanding functions, rates of change, and the behavior of graphs. When students see how these mathematical principles apply to real-world scenarios, engagement and comprehension can significantly increase. As educators, it's important to provide thoughtful explanations and innovative exercises that connect abstract mathematical ideas to tangible outcomes.