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Chapter 0: Problem 48

Adjust the graphing window to identify all vertical asympotes. $$f(x)=\frac{2 x}{\sqrt{x^{2}+x}}$$

Short Answer

Expert verified
The function has vertical asymptotes at \(x=0\) and \(x=-1\).

Step by step solution

01

Find critical values

The function \(f(x) = \frac{2x}{\sqrt{x^{2}+x}}\) would be undefined where the denominator, \(\sqrt{x^{2}+x}\), is equal to zero. To find these 'critical values', set \(x^{2} + x = 0\). This is a simple quadratic equation, which can be solved either by factoring or using the quadratic formula.
02

Solve for x

Solving for x from the equation in step 1, we get \(x=0\) or \(x=-1\). However, we need to confirm whether these points are vertical asymptotes or holes. A vertical asymptote occurs when the limit as x approaches that value is infinity or negative infinity.
03

Verify the limits at these points

This is done by finding the limit of the function as x approaches 0 from both left and right. Similarly, find the limit as x approaches -1 from both left and right using the limit definition. If any of these limits give us positive or negative infinity, then a vertical asymptote exists at that x-value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Critical Values
Critical values in mathematics are where a function's behavior changes significantly. For the function \(f(x) = \frac{2x}{\sqrt{x^{2}+x}}\), critical values occur where the denominator becomes zero, making the function undefined. To find these values, solve the equation \(x^{2} + x = 0\).
  • Factor the quadratic equation: \( x(x + 1) = 0 \).
  • Find solutions for \(x\): \(x = 0\) or \(x = -1\).
These solutions, \(x = 0\) and \(x = -1\), are where the function could potentially have vertical asymptotes or holes. Understanding these critical values is essential before further analysis.
Limit Definition and Vertical Asymptotes
The concept of a limit is central to identifying the behavior of functions as they approach certain critical values. A vertical asymptote is a line where the function's value grows indefinitely as the input gets close to a certain critical value. This can be tested using limits.For \(f(x) = \frac{2x}{\sqrt{x^{2}+x}}\), examine the behavior as \(x\) approaches 0 and -1. According to the limit definition:
  • If \( \lim_{x \to c^+} f(x) = \pm \infty \) or \( \lim_{x \to c^-} f(x) = \pm \infty \), then \(x = c\) is a vertical asymptote.
Plug \(c = 0\) and \(c = -1\) to check these limits. If any of these limits return infinite values, there is indeed a vertical asymptote at that point.
Solving Quadratic Equations
Quadratic equations are fundamental in determining critical points for functions. A basic form of a quadratic equation is \(ax^2 + bx + c = 0\). The solutions can be found using several methods:
  • Factoring: But only if the equation is easily factorable.
  • The Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
For the equation \(x^2 + x = 0\) we can use factoring since it is easily factorable:
  • Take out the greatest common factor: \(x(x + 1) = 0\).
  • Set each factor equal to zero: \(x = 0\) or \(x = -1\).
These techniques help efficiently find points where a function is not defined, assisting in understanding the function's behavior.
Graphing Functions with Vertical Asymptotes
Graphing functions can visually demonstrate where vertical asymptotes occur. A vertical asymptote represents a line where the graph heads towards positive or negative infinity.Start graphing the function \(f(x) = \frac{2x}{\sqrt{x^{2}+x}}\) by:
  • Identifying the critical points, such as \(x = 0\) and \(x = -1\).
  • Determining the behavior of the graph around these values by checking the limits to see if they become infinite.
Use graphing software or plot by hand, ensuring to adjust the window for a detailed view around critical values. This visual representation helps solidify understanding of how vertical asymptotes dictate the shape and behavior of graphs.

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