Problem 20 - Thomas Calculus

Question

Find the natural domain and graph the functions in Exercises \(15-20 .\) $$G(t)=1 /|t|$$

Step-by-Step Solution

Answer

The natural domain is \((-\infty, 0) \cup (0, \infty)\); graph has an asymptote at \(t=0\) and branches in quadrants I and II.

Step 1: Understanding the Function

The function given is \( G(t) = \frac{1}{|t|} \). This function involves an absolute value in the denominator. The absolute value of a number is always non-negative, meaning it is zero or positive.

Step 2: Determining Restricted Values

For the function \( G(t) = \frac{1}{|t|} \) to be defined, the denominator \( |t| \) must not be zero because division by zero is undefined. Thus, we require \( |t| eq 0 \), which implies that \( t eq 0 \).

Step 3: Natural Domain of the Function

The natural domain of \( G(t) = \frac{1}{|t|} \) consists of all real numbers except zero. Therefore, the natural domain is all real numbers \( \{t: t \in \mathbb{R}, t eq 0\} \), which can be written in interval notation as \(( -\infty, 0) \cup (0, \infty)\).

Step 4: Graphing the Function

To graph \( G(t) = \frac{1}{|t|} \), first note the function is undefined at \( t = 0 \). The graph will approach but never touch the vertical line \( t = 0 \), creating a vertical asymptote there. For \( t > 0 \), the graph is \( G(t) = \frac{1}{t} \). For \( t < 0 \), since \( |t| = -t \), the graph is also \( G(t) = \frac{1}{-t} \). Therefore, the function is symmetric with respect to the y-axis and will have branches in quadrants I and II that approach the y-axis without touching it.

Key Concepts

Absolute Value Function

An absolute value function is a function that involves the absolute value of a variable. The absolute value of a number is its distance from zero on the number line, without considering its direction. Hence, it is always non-negative. For the function we are considering,\( G(t) = \frac{1}{|t|} \), the absolute value appears in the denominator. This means that the function's behavior is influenced directly by the absolute value of \( t \).
\(|t|\) ensures that whether \( t \) is positive or negative, the output is non-negative. However, it is essential to note that absolute values can never be zero without making the expression undefined. This leads to an interesting element in defining the domain since \( G(t) \) cannot be evaluated when \( t = 0 \).

• The absolute value guarantees that the function outputs remain non-negative for values of \( t \) except at zero.
• This property removes the possibility of having zero in the denominator.

Keep in mind, while absolute values protect from negatives, they necessitate additional considerations in piecewise definitions and continuity, especially around zero or where special conditions apply.

Graphing Functions

To graph a function like \( G(t) = \frac{1}{|t|} \), it is beneficial to analyze the behavior of the function piece by piece. Observing the function's behavior for different segments of \( t \) helps in understanding how to graph it effectively.
First, acknowledge the function is undefined at \( t = 0 \). With this understanding, note that the graph will not intersect the vertical line where \( t = 0 \), indicating a vertical asymptote there. This guides our next steps in plotting the graph.

• For \( t > 0 \), the function simplifies to \( G(t) = \frac{1}{t} \), which is a simple rational function curve in the first quadrant.
• For \( t < 0 \), again because \(|t| = -t\), the function becomes \( G(t) = \frac{1}{-t} \), which also portrays a curve in the second quadrant.

These steps reveal that the branches of the graph are mirror images across the y-axis due to symmetry. Be sure to safeguard these characteristics and the undefined nature at zero while sketching the graph. Observing these pointers ensures an accurate and informative representation of \( G(t) \).

Vertical Asymptote

A vertical asymptote is a line that a graph will approach infinitely close but never actually touch. For the function \( G(t) = \frac{1}{|t|} \), a vertical asymptote exists at \( t = 0 \). This is because the function becomes undefined as it involves division by zero at this point, hence, \( t eq 0 \).
When graphing, this vertical asymptote at \( t = 0 \) indicates a form of infinite behavior as \( t \) nears zero from either direction.

• The branches of the graph approach the y-axis but never cross or meet it.
• For \( t > 0 \), the graph moves towards positive infinity as it approaches zero.
• For \( t < 0 \), the graph also trends towards positive infinity, displaying mirror symmetry.

The significance of vertical asymptotes lies in their depiction of function limits and unresolvable points. Their presence informs us that certain inputs cannot produce a valid output, hence structuring the domain and range of the function. Recognizing vertical asymptotes is vital when interpreting function behaviors and ensuring accurate graphical representation.