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Magazine Chapter 1: Problem 20
Find the natural domain and graph the functions. $$G(t)=1 /|t|$$
Short Answer
Expert verified
Domain is \((-\infty, 0) \cup (0, \infty)\). The graph has hyperbolic branches symmetric about the y-axis, with a vertical asymptote at \( t = 0 \).
Step by step solution
01
Understanding the Function
The given function is \( G(t) = \frac{1}{|t|} \). This function involves an absolute value and a division operation where the variable \( t \) is in the denominator.
02
Finding the Domain
To find the natural domain, we need to identify the values of \( t \) for which the function is defined. The function involves division by \(|t|\), which must not equal zero. Therefore, \( t \) cannot be 0, as the absolute value of 0 is zero, leading to division by zero.
03
Expressing the Domain
From Step 2, we know that the function \( G(t) \) is undefined at \( t = 0 \). Therefore, the natural domain consists of all real numbers except 0. The domain can be expressed as: \( \{ t \in \mathbb{R} \,|\, t eq 0 \} \) or in interval notation as: \((-\infty, 0) \cup (0, \infty)\).
04
Graphing the Function
When graphing \( G(t) = \frac{1}{|t|} \), notice two regions: one where \( t > 0 \) and another where \( t < 0 \). For \( t > 0 \), \( |t| = t \), making the graph \( G(t) = \frac{1}{t} \) (similar to the basic reciprocal function). For \( t < 0 \), \( |t| = -t \), so the graph is also \( G(t) = -\frac{1}{t} \). This results in two separate branches that are symmetrical with respect to the y-axis (or even function).
05
Final Graph Interpretation
The graph of \( G(t) \) has two hyperbolic branches: one in quadrants I and IV where \( t > 0 \) and the other in quadrants II and III where \( t < 0 \). Each branch has a vertical asymptote at \( t = 0 \) and approaches the y-axis vertically without touching.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value
The absolute value of a number is its distance from zero on a number line and is always non-negative. In simpler terms, it tells us how far a number is from zero, without considering its direction (positive or negative). When dealing with the function \( G(t) = \frac{1}{|t|} \), the absolute value affects how we perceive the variable \( t \).
- If \( t > 0 \), the absolute value \(|t| = t\).
- If \( t < 0 \), the absolute value \(|t| = -t\).
Natural Domain
The natural domain of a function includes all the possible values that the input, or independent variable, can take, resulting in real output values. In the function \( G(t) = \frac{1}{|t|} \), we deal with division by the expression \( |t| \), which cannot be zero. This is because dividing by zero is undefined in mathematics.Therefore, the natural domain of \( G(t) \) excludes the point where \( t = 0 \). The natural domain can be expressed as:
- Set Notation: \( \{ t \in \mathbb{R} \,|\, t eq 0 \} \)
- Interval Notation: \( (-\infty, 0) \cup (0, \infty) \)
Graph Interpretation
Graphing involves visually representing a function to understand its behavior. For \( G(t) = \frac{1}{|t|} \), the function has distinct behaviors based on the value of \( t \).
- For \( t > 0 \), the graph resembles \( \frac{1}{t} \), a standard reciprocal function that decreases as \( t \) increases.
- For \( t < 0 \), the graph resembles \( -\frac{1}{t} \), effectively mirroring the positive side about the y-axis.
Reciprocal Function
A reciprocal function is a type of function that involves the reciprocal of a variable, typically written as \( \frac{1}{x} \). It is characterized by hyperbolic branches and a vertical asymptote at the point where the function is undefined. In the case of \( G(t) = \frac{1}{|t|} \), the reciprocal nature of the function is influenced by the absolute value.Key characteristics:
- Two symmetric branches lie in opposite quadrants.
- Both branches approach the vertical asymptote \( t = 0 \) as they tend to positive or negative infinity.
