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Chapter 2: Problem 55

Find all asymptotes of the graph of the given equation. $$ y=32 /\left(8-2^{x}\right) $$

Short Answer

Expert verified
Vertical asymptote at \( x = 3 \); horizontal asymptote at \( y = 0 \) as \( x \to \infty \) and \( y = 4 \) as \( x \to -\infty \).

Step by step solution

01

Identify Vertical Asymptotes

A vertical asymptote occurs where the denominator of the function is zero but the numerator isn't. For the function \( y = \frac{32}{8 - 2^x} \), set the denominator equal to zero: \[ 8 - 2^x = 0 \]This gives us \[ 2^x = 8 \]We can write 8 as a power of 2: \[ 2^x = 2^3 \]By equating the exponents, we find \[ x = 3 \]Thus, there is a vertical asymptote at \( x = 3 \).
02

Identify Horizontal Asymptotes

Horizontal asymptotes occur based on the behavior of \( y \) as \( x \) approaches either infinity or negative infinity. For large \( x \), the denominator \( 8 - 2^x \) becomes very large and negative, thus\[ y = \frac{32}{8 - 2^x} \to 0 \text{ as } x \to \infty \]As \( x \to -\infty \), \( 2^x \to 0 \), so\[ y = \frac{32}{8 - 0} = 4 \]So, there is a horizontal asymptote at \( y = 0 \) as \( x \to \infty \) and \( y = 4 \) as \( x \to -\infty \).

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Key Concepts

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

Vertical Asymptotes
Vertical asymptotes are imaginary lines that the graph of a function approaches as the independent variable approaches a certain value. But the function never actually touches or crosses these lines. They occur in the context of rational functions when the denominator becomes zero but the numerator remains non-zero, leading to division by zero. This results in the value of the function shooting up to infinity or dropping down to negative infinity.

To find vertical asymptotes in the given function, we focus on the denominator:
  • The function is: \( y= \frac{32}{8 - 2^x} \).
  • Set the denominator equal to zero: \( 8 - 2^x = 0 \).
  • Solve for \( x \) by expressing the equation in terms of powers of 2: \( 2^x = 8 \), which simplifies to \( 2^3 \).
  • Equating the exponents gives \( x = 3 \).
Therefore, there is a vertical asymptote at \( x = 3 \). This is where the function shoots off to infinity or negative infinity.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a graph as the independent variable either increases or decreases without bound. For a function \( y = f(x) \), horizontal asymptotes occur if \( y \) approaches a constant value as \( x \) approaches infinity or negative infinity.

In the exercise, examine the horizontal asymptotes:
  • Start by considering \( x \to \infty \): The denominator \( 8 - 2^x \) becomes large and negative due to the exponential growth of \( 2^x \).
  • Therefore, \( y \to 0 \) because the fraction's numerator (32) remains constant while the denominator grows infinitely large.
  • Consider \( x \to -\infty \): In this case, \( 2^x \to 0 \), so the denominator is simply \( 8 \).
  • Thus, \( y = \frac{32}{8} = 4 \).
Hence, the function has a horizontal asymptote at \( y = 0 \) as \( x \to \infty \) and another at \( y = 4 \) as \( x \to -\infty \). This tells us how the function behaves as it moves far away in both directions.
Exponential Functions
Exponential functions are a specific type of mathematical function where a constant base is raised to a variable exponent, written as \( a^x \). These functions are central to the concepts of growth and decay across various fields like biology, finance, and physics.

In this exercise, the exponential function is \( 2^x \). Here鈥檚 why it鈥檚 significant:
  • Exponential growth: As \( x \) increases, \( 2^x \) grows very quickly, leading to the denominator \( 8 - 2^x \) becoming negative and large, influencing the behavior of the graph as \( x \to \infty \).
  • Exponential decay: As \( x \) approaches negative infinity, \( 2^x \to 0 \). This causes the denominator to stabilize at \( 8 \), simplifying the fraction and affecting the horizontal asymptote \( y = 4 \).
  • Impact on asymptotes: Changes in \( 2^x \) ultimately determine both vertical and horizontal asymptotes, shaping the entire graph of the function.
Understanding how exponential functions like \( 2^x \) behave helps predict how other similar functions might behave and assists in visualizing the entire graph. They dramatically affect how the function approaches its asymptotes.

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Most popular questions from this chapter

An assertion is made about a function \(f\) that is defined on a closed, bounded interval. If the statement is true, explain why. Otherwise, sketch a function \(f\) that shows it is false. (Note: \(|f|\) is defined by \(|f|(x)=|f(x)| .)\) If \(f\) is continuous, then \(f^{2}\) is continuous.

Use the Intermediate Value Theorem to show that \(x / 2=\sin (x)\) has a positive solution. (Consider the function \(f(x)=x / 2-\sin (x)\) and the points \(a=\pi / 6\) and \(b=2.1 .)\)

When a drug is intravenously introduced into a patient's bloodstream at a constant rate, the concentration \(C\) of the drug in the patient's body is typically given by $$ C(t)=\frac{\alpha}{\beta}\left(1-e^{-\beta t}\right) $$ where \(\alpha\) and \(\beta\) are positive constants. What is the limiting concentration \(\lim _{t \rightarrow \infty} C(t) ?\) Sketch the graph of \(C .\) What horizontal asymptote does it have?

In Exercises \(67-73,\) use algebraic manipulation (as in Example 5 ) to evaluate the limit. $$ \lim _{x \rightarrow 4}\left(\frac{\sqrt{8-x-2}}{\sqrt{5-x-1}}\right) $$

In Exercises \(83-86,\) use the Pinching Theorem to establish the required limit. $$ \begin{aligned} &\text { If }|g(x)-f(x)| \leq \sqrt{|x|}, \text { and if } \lim _{x \rightarrow 0} f(x)=\ell, \text { then }\\\ &\lim _{x \rightarrow 0} g(x)=\ell \end{aligned} $$
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