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Chapter 4: Problem 30

$$ F(x)=\left\\{\begin{array}{lc} 0 & x<-2 \\ 0.25 x+0.5 & -2 \leq x<1 \\ 0.5 x+0.25 & 1 \leq x<1.5 \\ 1 & 1.5 \leq x \end{array}\right. $$

Short Answer

Expert verified
Evaluate the function using its respective formula based on the value of \( x \) with four defined pieces.

Step by step solution

01

Identify the Function Piece

Before evaluating the function at any specific point, identify which piece of the piecewise function will be used for a given value of \(x\). The function has specific expressions depending on the range of \(x\).
02

Define Function Pieces

The piecewise function \( F(x) \) has four pieces: 1. \( F(x) = 0 \) for \( x < -2 \)2. \( F(x) = 0.25x + 0.5 \) for \( -2 \leq x < 1 \)3. \( F(x) = 0.5x + 0.25 \) for \( 1 \leq x < 1.5 \)4. \( F(x) = 1 \) for \( x \geq 1.5 \)
03

Evaluate Each Case

To understand how to work with this function, evaluate different cases:- For \( x = -3 \), use \( F(x) = 0 \)- For \( x = 0 \), use \( F(x) = 0.25(0) + 0.5 = 0.5 \)- For \( x = 1.2 \), use \( F(x) = 0.5(1.2) + 0.25 = 0.85 \)- For \( x = 2 \), use \( F(x) = 1 \)
04

Generalize Solution

Recall the breakdown of ranges and formulas to evaluate \( F(x) \) for any \( x \). Each piece adheres to its conditions based on the value of \( x \). Using these rules allows solving for any \( F(x) \) scenario.

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Key Concepts

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

Evaluating Functions
Evaluating piecewise functions involves finding the value of the function based on the specific interval that the input, or \( x \), falls into. Each "piece" of the function applies to a distinct range of \( x \) values. For instance, in the given function \( F(x) \), there are four separate expressions, depending on the range of \( x \).
  • When \( x < -2 \), the value of \( F(x) \) is 0.
  • For \( -2 \leq x < 1 \), use the expression \( 0.25x + 0.5 \).
  • If \( 1 \leq x < 1.5 \), apply \( 0.5x + 0.25 \).
  • Finally, when \( x \geq 1.5 \), \( F(x) \) is always 1.
Breaking down each part of the piecewise function helps you determine the appropriate expression to use based on the input value. Evaluate by substituting \( x \) into this expression.
Function Ranges
Understanding the function ranges in a piecewise function is crucial for determining which expression to use. A piecewise function is defined by different formulas over specific intervals, which are the function ranges. Here's how they work in \( F(x) \):
  • The first range is \( x < -2 \), where the function is simply a constant, \( 0 \).
  • From \( x = -2 \) to just under \( x = 1 \), the linear function \( 0.25x + 0.5 \) is used, indicating a gradual increase as \( x \) increases.
  • Between \( x = 1 \) and \( x = 1.5 \), \( F(x) \) is determined by the line \( 0.5x + 0.25 \), which grows faster than the previous interval due to the higher coefficient of \( x \).
  • At \( x \geq 1.5 \), the function simplifies to a constant value of \( 1 \), regardless of how large \( x \) becomes.
Each range specifies different behavior of the function, making it indispensable to recognize which formula applies based on the value of \( x \).
Mathematical Piecewise Expressions
Mathematical piecewise expressions allow functions to be defined differently across various intervals of the input variable. This structure makes piecewise functions extremely versatile for modeling situations where behavior changes.In this function, notice how the various expressions define different linear behaviors and constants in distinct intervals.
  • The constant expressions \( F(x) = 0 \) and \( F(x) = 1 \) depict horizontal lines across specific intervals.
  • The linear expressions \( 0.25x + 0.5 \) and \( 0.5x + 0.25 \) showcase straight lines where the slope determines the rate at which \( F(x) \) increases with \( x \).
The piecewise function can represent complex real-world scenarios where changes occur in steps, such as pricing schemes, tax brackets, or physical phenomena. Understanding how to interpret and construct these functions is vital for mathematical modeling and analysis.

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Most popular questions from this chapter

An article in Applied Mathematics and Computation ["Confidence Intervals for Steady State Availability of a System with Exponential Operating Time and Lognormal Repair Time" \((2003,\) Vol. \(137(2),\) pp. \(499-509)]\) considered the long-run availability of a system with an assumed lognormal distribution for repair time. In a given example, repair time follows a lognormal distribution with \(\theta=\omega=1 .\) Determine the following: (a) Probability that repair time is more than five time units (b) Conditional probability that a repair time is less than eight time units given that it is more than five time units (c) Mean and variance of repair time

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When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each caller orders one ticket. What is the probability that the bus is filled in less than three hours from the time of the fare reduction?

Without an automated irrigation system, the height of plants two weeks after germination is normally distributed with a mean of 2.5 centimeters and a standard deviation of 0.5 centimeter. (a) What is the probability that a plant's height is greater than 2.25 centimeters? (b) What is the probability that a plant's height is between 2.0 and 3.0 centimeters?
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