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Chapter 4: Problem 29

$$ F(x)=\left\\{\begin{array}{lr} 0 & x<0 \\ 0.2 x & 0 \leq x<4 \\ 0.04 x+0.64 & 4 \leq x<9 \\ 1 & 9 \leq x \end{array}\right. $$

Short Answer

Expert verified
The function \( F(x) \) changes based on intervals: 0 for \( x < 0 \), linear for \( 0 \leq x < 4 \) and \( 4 \leq x < 9 \), and constant (1) for \( x \geq 9 \).

Step by step solution

01

Understanding Piecewise Function

The function \( F(x) \) is defined in pieces depending on the value of \( x \). Each piece has its own expression based on the range of \( x \). You need to evaluate these separately by plugging in specific values of \( x \) to understand how the function behaves across different intervals.
02

Evaluate for \( x < 0 \)

In this range, \( F(x) = 0 \). This means that no matter what negative value you choose for \( x \), the function value will always be 0.
03

Evaluate for \( 0 \leq x < 4 \)

For values of \( x \) between 0 and 4, the function is \( F(x) = 0.2x \). For example, if \( x = 2 \), then \( F(2) = 0.2 \times 2 = 0.4 \).
04

Evaluate for \( 4 \leq x < 9 \)

In this interval, the expression is \( F(x) = 0.04x + 0.64 \). For example, if \( x = 5 \), then \( F(5) = 0.04 \times 5 + 0.64 = 0.84 \).
05

Evaluate for \( x \geq 9 \)

Here, the function is constant and \( F(x) = 1 \). This means for any \( x \) that is 9 or greater, the function value is always 1.

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Key Concepts

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

Piecewise Function
A piecewise function is a kind of mathematical function where different formulas or expressions are used to calculate the output based on the input value. This means that the formula you use depends on which interval the input falls into. For the function given,
  • When the input, or value of \( x \), is less than 0, the output is always 0.
  • If \( x \) is between 0 and 4, the expression \( 0.2x \) is used.
  • In the interval from 4 to 9, the function is given by \( 0.04x + 0.64 \).
  • Finally, for \( x \) values that are 9 or greater, the function value is always 1.
Defining functions in pieces like this allows us to model complex behavior, where the output doesn't follow a single formula across all input values.
Piecewise functions are incredibly useful in real-life applications where a situation or process changes behavior at certain thresholds.
Function Evaluation
Function evaluation refers to the process of determining the output of a function given an input. In the context of piecewise functions, this involves identifying which piece of the function to use based on the input value.
This function evaluation process can be thought of as testing the value of \( x \) to see which part of the function applies.
  • For negative \( x \), the constant 0 is used.
  • Positive inputs up to 4 require the calculation of \( 0.2x \).
  • If \( x \) is between 4 and 9, you calculate \( 0.04x + 0.64 \).
  • Values 9 and higher mean the function evaluates to 1, no further calculation needed.
      • Each evaluation often involves simple arithmetic, making it approachable once the correct expression is chosen for the specific interval where \( x \) falls. Understanding how to evaluate functions effectively is essential for tackling more complex mathematical challenges.
Step-by-Step Solution
Solving piecewise functions step-by-step ensures clarity in understanding each segment of the process. Let's walk through the solution using our function:
  • **Step 1**: Recognize that since it's a piecewise function, start by noting where your \( x \) value fits in the range boundaries.
  • **Step 2**: For \( x < 0 \), simply set \( F(x) = 0 \).
  • **Step 3**: For \( 0 \leq x < 4 \), use \( F(x) = 0.2x \). For example, \( x = 2 \) leads to \( 0.2 \times 2 = 0.4 \).
  • **Step 4**: When \( 4 \leq x < 9 \), apply \( F(x) = 0.04x + 0.64 \). For instance, \( x = 5 \) computes to \( 0.04 \times 5 + 0.64 = 0.84 \).
  • **Step 5**: For \( x \geq 9 \), the function is set at \( F(x) = 1 \). Each step requires careful assessment of the interval to apply the correct function piece, ensuring accuracy by following through the arithmetic as shown in the examples.
Breaking down the process into steps helps demystify how each segment of the piecewise function contributes to the overall output.

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