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Chapter 1: Problem 20

Find the natural domain and graph the functions. \(G(t)=1 /|t|\)

Short Answer

Expert verified
The domain of \( G(t) = \frac{1}{|t|} \) is \(t \in (-\infty, 0) \cup (0, \infty)\). The graph has a vertical asymptote at \(t = 0\).

Step by step solution

01

Identify the Expressions Involved in the Function

The function we have is \( G(t) = \frac{1}{|t|} \). This function involves an absolute value, \(|t|\), and a division by that absolute value, which we need to consider while finding the domain.
02

Find the Domain Restrictions

The function \( G(t) = \frac{1}{|t|} \) is undefined where the denominator is zero, since division by zero is undefined. Therefore, we must solve \(|t| = 0\). This occurs when \(t = 0\).
03

Determine the Natural Domain

Since \(|t| eq 0\) and \(t = 0\) is the point where \(|t|\) becomes 0, the natural domain of the function excludes \(t = 0\). Hence, the natural domain is all real numbers except 0, which is expressed as \(t \in (-\infty, 0) \cup (0, \infty)\).
04

Discuss the Graph of the Function

The function \( G(t) = \frac{1}{|t|} \) is undefined at \(t = 0\), so the graph will have a vertical asymptote there. For \(t > 0\), \(|t| = t\) and the function becomes \( \frac{1}{t} \). For \(t < 0\), \(|t| = -t\) and the function again becomes \( \frac{1}{-t} \) or \( -\frac{1}{t} \). The graph will approach 0 as \(t\) approaches infinity in both the positive and negative directions, resembling a hyperbola.

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Key Concepts

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

Understanding the Absolute Value Function
The absolute value function is essential in mathematics, representing the distance of a number from zero on the number line. For any real number \(t\), the absolute value is denoted by \(|t|\) and is always non-negative:
  • \(|t| = t\) if \(t \geq 0\),
  • \(|t| = -t\) if \(t < 0\).
This concept is vital in the function \(G(t) = \frac{1}{|t|}\), where it dictates how the function behaves differently based on the sign of \(t\). Remembering that \(|t|\) never results in a negative value helps predict and graph functions involving absolute values.
Problems with Division by Zero
One critical restriction when dealing with functions is avoiding division by zero. In mathematics, division by zero is undefined because it leads to contradictions and operations that do not make sense. For example, if we try to divide 1 by 0, there is no number that can satisfy the equation.
In the function \(G(t) = \frac{1}{|t|}\), division by zero occurs when \( |t| = 0 \), which happens when \( t = 0 \). Therefore, this point must be excluded from the domain, ensuring that the function remains valid across its natural domain.
The Importance of Vertical Asymptotes
Vertical asymptotes appear in the graph of a function where the function approaches infinity. These occur due to division by zero or limits approaching a non-finite value.
  • In the function \(G(t) = \frac{1}{|t|}\), the vertical asymptote is at \(t = 0\).
  • This means as \(t\) gets closer to zero from either side, the value of the function increases without bound.
This forms a critical feature of the graph, creating a division in the function's existence and often illustrating approaches toward infinity or negative infinity.
Exploring Real Numbers
Real numbers encompass all the numbers on the continuous number line, including rational and irrational numbers. It's the set of numbers that excludes imaginary numbers and is often represented by the symbol \( \mathbb{R} \).
When we talk about the natural domain of a function such as \(G(t) = \frac{1}{|t|}\), we are referencing all real numbers except for those that cause problems like division by zero. In this case, the domain is all real numbers except \(t = 0\), or symbolically \(t \in (-\infty, 0) \cup (0, \infty)\).
  • The comprehensive nature of real numbers makes them deeply applicable in most mathematical scenarios, underscoring their significance when discussing function domains.

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Most popular questions from this chapter

In Exercises \(71-74,\) you will explore graphically the general sine function $$f(x)=A \sin \left(\frac{2 \pi}{B}(x-C)\right)+D$$ as you change the values of the constants \(A, B, C,\) and \(D .\) Use a CAS or computer grapher to perform the steps in the exercises. The horizontal shift \(C\) Set the constants \(A=3, B=6, D=0\) . a. Plot \(f(x)\) for the values \(C=0,1,\) and 2 over the interval \(-4 \pi \leq x \leq 4 \pi .\) Describe what happens to the graph of the general sine function as \(C\) increases through positive values. b. What happens to the graph for negative values of \(C ?\) c. What smallest positive value should be assigned to \(C\) so the graph exhibits no horizontal shift? Confirm your answer with a plot.

Exercises \(59-68\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=1+\frac{1}{x^{2}}, \quad \text { compressed vertically by a factor of } 2 $$

Graph \(y=\sin x\) and \(y=\lfloor\sin x\rfloor\) together. What are the domain and range of \(\lfloor\sin x\rfloor ?\)

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side \(x\) at each corner and then folding up the sides as in the figure. Express the volume \(V\) of the box as a function of \(x .\)

Exercises \(59-68\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=\sqrt{x+1}, \quad \text { compressed horizontally by a factor of } 4 $$
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