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Chapter 0: Problem 21

Sketch the graph of \(F(t)=\frac{|t|-t}{t}\)

Short Answer

Expert verified
For \(t > 0\), \(F(t) = 0\), and for \(t < 0\), \(F(t) = -2\). The graph has a discontinuity at \(t = 0\).

Step by step solution

01

Identify the Function Components

The function is given as \( F(t) = \frac{|t| - t}{t} \). Note that it consists of an absolute value \(|t|\) and the variable \(t\) in both the numerator and the denominator.
02

Analyze the Function for Different Intervals

The expression \(|t|\) behaves differently based on the sign of \(t\). Let's consider two cases: when \(t > 0\) and when \(t < 0\).
03

Evaluate the Case \(t > 0\)

For \(t > 0\), \(|t| = t\). Hence, \( F(t) = \frac{|t| - t}{t} = \frac{t - t}{t} = 0\). So, for all \(t > 0\), \(F(t) = 0\).
04

Evaluate the Case \(t < 0\)

For \(t < 0\), \(|t| = -t\). Hence, \( F(t) = \frac{|t| - t}{t} = \frac{-t - t}{t} = \frac{-2t}{t} = -2\). Thus, for all \(t < 0\), \(F(t) = -2\).
05

Consider the Case \(t = 0\)

The function includes \(t\) in the denominator, which means \(F(t)\) is undefined at \(t = 0\).
06

Sketch the Graph

Combine the results of the previous steps to sketch the graph. For \(t > 0\), the graph is a horizontal line at \(y = 0\). For \(t < 0\), the graph is a horizontal line at \(y = -2\). The graph is not defined at \(t = 0\), indicating a point of discontinuity.

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Key Concepts

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

Absolute Value Function
The absolute value function is a fundamental mathematical concept that measures the distance of a number from zero on the number line. For any real number, the absolute value is always non-negative. It is represented by two vertical bars, such as \(|x|\). The key idea is:
  • When \(x > 0\), \(|x| = x\).
  • When \(x < 0\), \(|x| = -x\).
  • When \(x = 0\), \(|x| = 0\).
This function helps in treating positive and negative values equally, essentially 'ignoring' the sign. In the context of our exercise with the function \(F(t)=\frac{|t|-t}{t}\), the behavior of the absolute value changes the result based on whether \(t\) is positive, negative, or zero. This function's unique property allows us to split our analysis into different cases depending on the sign of \(t\).
Piecewise Function
A piecewise function is one that is defined by different expressions based on the value of the input. It is like piecing together different functions to create a whole. These functions are especially useful when a formula or a rule needs to reflect more than one behavior.
In our problem with \(F(t)=\frac{|t|-t}{t}\), we can break it down as follows:
  • For \(t > 0\), the function simplifies to \(F(t) = 0\).
  • For \(t < 0\), the function simplifies to \(F(t) = -2\).
  • At \(t = 0\), the function is undefined because division by zero is not possible.
The piecewise nature of \(F(t)\) allows us to easily evaluate and sketch its graph by considering each interval separately. This organized break-down sets the stage for a clearer understanding of the function's behavior across different domains.
Graph Discontinuity
Graph discontinuity occurs when there are breaks or jumps in the graph of a function. It happens at points where the function is not continuous. Understanding discontinuity is crucial, as it often highlights important features of the function.
In the function \(F(t)=\frac{|t|-t}{t}\), discontinuity is present at \(t = 0\). Here's why:
  • The function is not defined at \(t = 0\) because it leads to division by zero.
  • For \(t > 0\), the function is a horizontal line at \(y = 0\).
  • For \(t < 0\), the function is a different horizontal line at \(y = -2\).
The shift from one horizontal line to another, with an undefined point at \(t = 0\), marks a discontinuity. Graphically, this would be observed as a break or a gap on the graph between these two lines. This discontinuity is essential for understanding where the function fails to be seamless and how it behaves in distinct intervals.

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Most popular questions from this chapter

Write \(p(x)=1 / \sqrt{x^{2}+1}\) as a composite of four functions.

Suppose that \(f\) is an even function satisfying \(f(x)=-1+\sqrt{x}\) for \(x \geq 0\). Sketch the graph of \(f\) for \(-4 \leq x \leq 4\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{2}-y^{2}=4\)

Your computer algebra system (CAS) may allow the use of parameters in defining functions. In each case, draw the graph of \(y=f(x)\) for the specified values of the parameter \(k\), using the same axes and \(-5 \leq x \leq 5\). (a) \(f(x)=|k x|^{0.7}\) for \(k=1,2,0.5,\) and 0.2 . (b) \(f(x)=|x-k|^{0.7}\) for \(k=0,2,-0.5,\) and \(-3 .\) (c) \(f(x)=|x|^{k}\) for \(k=0.4,0.7,1,\) and \(1.7 .\)

Sketch the graph of each equation. $$ x=y^{2}-3 $$
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