Discontinuity At X=7: Find The Rational Function W(x)

Nov 4, 2025 by Admin 54 views

Let's dive into the fascinating world of rational functions and discontinuities! This question explores how horizontal shifts affect the points of discontinuity in these functions. We'll break down the concept, analyze the options, and arrive at the correct answer, making sure you understand the why behind it all. So, if you've ever wondered how transformations impact the behavior of functions, you're in the right place. Let's get started, guys!

Understanding Points of Discontinuity

First off, what exactly is a point of discontinuity? In the realm of rational functions (functions that are ratios of polynomials), these points pop up where the denominator equals zero. Imagine dividing by zero – it's a mathematical no-no, creating a break or "discontinuity" in the graph. These discontinuities can manifest as vertical asymptotes (where the function shoots off to infinity) or holes (where there's a removable gap). Identifying these points is crucial for understanding the function's overall behavior. Think of it like finding the potholes on a road map – you need to know where they are to navigate smoothly. In the context of our problem, both rational functions v(x) and w(x) have a discontinuity at x = 7. This means that if you were to plug x = 7 into the denominator of either function (in its simplified form), you'd get zero. This is the key piece of information we'll use to solve the puzzle.

Why does the denominator being zero cause a discontinuity? Well, division by zero is undefined in mathematics. When the denominator of a rational function approaches zero, the overall value of the function either skyrockets towards infinity (creating a vertical asymptote) or becomes undefined at a specific point (creating a hole). This "undefinedness" is what we call a discontinuity. Now that we understand what a point of discontinuity is and why it occurs, we can delve into how transformations affect these points. Specifically, we're interested in horizontal shifts and how they can move the location of discontinuities on the graph. This is the core concept we need to grasp to tackle the problem at hand, so let's explore it further.

Analyzing Horizontal Shifts

The crux of this problem lies in understanding how horizontal shifts affect a function's graph. A horizontal shift occurs when we replace x with (x - c) or (x + c) within the function's expression. The key takeaway here is that replacing x with (x - c) shifts the graph to the right by c units, while replacing x with (x + c) shifts the graph to the left by c units. This might seem counterintuitive at first, but it's a fundamental concept in function transformations. To illustrate this, let's consider a simple function like f(x) = 1/x, which has a vertical asymptote (a point of discontinuity) at x = 0. If we replace x with (x - 2), we get f(x - 2) = 1/(x - 2). The vertical asymptote has now shifted from x = 0 to x = 2, a shift of 2 units to the right. Conversely, if we replace x with (x + 2), we get f(x + 2) = 1/(x + 2), and the vertical asymptote shifts to x = -2, a shift of 2 units to the left.

This principle applies to any point on the graph, including points of discontinuity. So, if v(x) has a discontinuity at x = 7, then v(x - 7) will have a discontinuity at x = 14 (a shift of 7 units to the right), and v(x + 7) will have a discontinuity at x = 0 (a shift of 7 units to the left). Understanding this horizontal shift is crucial for pinpointing the correct equation for w(x). We need to find the transformation that maintains the discontinuity at x = 7. Vertical shifts, on the other hand, do not affect the location of discontinuities. Adding a constant to the function (like v(x) + 7) simply moves the entire graph up or down without changing the x-values where the discontinuities occur. Now, with a solid understanding of horizontal shifts, let's dissect the given options and see which one fits the bill.

Evaluating the Options

Now, let's break down each option to see which one could represent w(x), keeping in mind that w(x) also has a point of discontinuity at x = 7.

A. w(x) = v(x - 7): This option represents a horizontal shift. We're replacing x with (x - 7) inside the function v. As we discussed earlier, this shifts the graph of v(x) seven units to the right. If v(x) has a discontinuity at x = 7, then v(x - 7) will have a discontinuity at x = 14. This is because plugging in x = 14 into (x - 7) gives us 7, which is where v(x) has its discontinuity. Therefore, this option is incorrect because it shifts the discontinuity away from x = 7.
B. w(x) = v(x + 7): This option also represents a horizontal shift, but this time we're replacing x with (x + 7). This shifts the graph of v(x) seven units to the left. If v(x) has a discontinuity at x = 7, then v(x + 7) will have a discontinuity at x = 0. Plugging in x = 0 into (x + 7) gives us 7, which is where v(x) has its discontinuity. So, this option is also incorrect as it moves the discontinuity away from x = 7.
C. w(x) = v(x - 7) + 7: This option combines a horizontal shift and a vertical shift. The (x - 7) part shifts the graph seven units to the right, moving the discontinuity away from x = 7 (as we saw in option A). The + 7 part shifts the entire graph upwards by 7 units, but this vertical shift doesn't affect the x-coordinate of the discontinuity. Therefore, this option is incorrect because the horizontal shift moves the discontinuity.
D. w(x) = v(x) + 7: This option represents a vertical shift. We're simply adding 7 to the entire function v(x). This shifts the graph upwards by 7 units. A vertical shift does not affect the x-coordinate of any points on the graph, including points of discontinuity. If v(x) has a discontinuity at x = 7, then v(x) + 7 will also have a discontinuity at x = 7. The vertical shift only changes the y-values, not the x-values where the discontinuities occur. Thus, this is the correct option.

The Correct Answer

Based on our analysis, the only equation that could represent w(x), maintaining the discontinuity at x = 7, is D. w(x) = v(x) + 7. The vertical shift doesn't alter the location of the discontinuity, making it the perfect fit. Understanding the impact of horizontal and vertical shifts is key to navigating these types of problems. So, next time you encounter a function transformation question, remember to consider how each transformation affects the graph's key features, like discontinuities. You've got this, guys!