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Chapter 1: Problem 6

What is a piecewise linear function?

Short Answer

Expert verified
Answer: A piecewise linear function is a function defined by multiple linear functions on different intervals of the input variable. To create a piecewise linear function, follow these steps: 1) Determine the intervals for the different linear functions. 2) Define a separate linear function (of the form f(x) = ax + b) for each interval with different constants a and b. 3) Combine the defined functions into a single function, specifying the intervals for each function.

Step by step solution

01

Define a piecewise linear function

A piecewise linear function is a function that is defined by multiple linear functions on different intervals of the input variable. This means that the function can have different linear equations for different ranges of the input values, which are stitched together to form a single function.
02

Determine the intervals

To create a piecewise linear function, the first step is to determine the intervals on which the different linear functions will be defined. These intervals can be of any length and can be overlapping or non-overlapping. For example, consider the following intervals: \newline 1) \(x<0\) \newline 2) \(0 \leq x < 4\) \newline 3) \(x \geq 4\)
03

Define the linear functions

Now, for each interval, we will define a separate linear function. A linear function has the general form of \(f(x) = ax + b\), where a and b are constants. For each interval, we will choose different a and b values to define the linear functions. For example: \newline 1) \(f(x) = x + 2\) for \(x<0\) \newline 2) \(f(x) = -x + 4\) for \(0 \leq x < 4\) \newline 3) \(f(x) = \frac{1}{2}x + 2\) for \(x \geq 4\)
04

Combine the functions

Finally, we will combine these functions to form the piecewise linear function. We will write the combined function as follows: \newline $ f(x) = \left\{ \begin{array}{ll} x + 2 & \mbox{if } x < 0 \\ -x + 4 & \mbox{if } 0 \leq x < 4 \\ \frac{1}{2}x + 2 & \mbox{if } x \geq 4 \end{array} \right. $ This piecewise linear function is now defined for all real numbers and consists of three different linear functions on their respective intervals.

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Key Concepts

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

Linear Equations
Linear equations are mathematical expressions that model straight-line relationships. The general formula for a linear equation is: \( y = ax + b \). Here, \( a \) is the slope of the line, which indicates how steep the line is. The value \( b \) is the y-intercept; it's where the line crosses the y-axis. Linear equations are essential because they simplify the complex relationships into manageable predictions.

Some characteristics of linear equations include:
  • Constant slope: This means the line maintains a uniform angle throughout its length.
  • Proportional relationship: Changes in one variable result in proportional changes in another.
  • Roots or zeros: Solutions where the line crosses the x-axis, i.e., \( y = 0 \).
Linear equations are the building blocks for more complex functions, like piecewise linear functions.
Interval Notation
Interval notation is a way of representing subsets of the real number line. It provides a concise way to express intervals where a function or certain mathematical conditions apply.

When creating a piecewise linear function, each segment of the function is defined over specific intervals. Interval notation uses brackets and parentheses to describe these segments:
  • (a, b): The interval includes all numbers between \( a \) and \( b \), not including \( a \) and \( b \) themselves.
  • [a, b): The interval includes \( a \) and all numbers up to \( b \), but not \( b \) itself.
  • [a, b]: This includes both \( a \) and \( b \).
  • (-鈭�, b): Represents all numbers less than \( b \).
Understanding interval notation is crucial when dealing with piecewise functions which operate differently across the number line.
Function Definition
A function is a fundamental concept in mathematics that describes a relationship between input values and output values, typically expressed as \( f(x) \). For any given input \( x \), the function provides exactly one output \( f(x) \).

In piecewise linear functions, the function is divided into segments, each with its own linear expression. Each segment corresponds to a specific interval. The function definition itself expresses how these intervals come together:
  • Identify intervals where each part of the function applies.
  • Apply the specific linear equation for each interval.
  • Write out the piecewise function with all segments clearly defined.
A piecewise function is fully defined only when each segment and its corresponding interval are specified.
Mathematical Continuity
Continuity in math refers to the property of a function where it is continuous at every point in its domain. For a function to be continuous, it cannot have any breaks, jumps, or holes.

Piecewise linear functions can be continuous, but they must be carefully constructed to ensure that the "pieces" fit together seamlessly where the intervals meet.

Key points for ensuring continuity:
  • The end of one interval should meet the beginning of the next without any gap.
  • Check that the right-hand limit equals the left-hand limit at each interval junction.
  • Sometimes, you need to adjust the formulas slightly to get smooth transitions between segments.
In the context of piecewise functions, ensuring a continuous transition where segments meet can make the difference between a smooth and a disjointed function.

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Most popular questions from this chapter

Amplitude and period Identify the amplitude and period of the following functions. $$q(x)=3.6 \cos (\pi x / 24)$$

Reciprocal bases Assume that \(b>0\) and \(b \neq 1 .\) Show that \(\log _{1 / b} x=-\log _{b} x\)

Determine whether the following statements are true and give an explanation or counterexample. a. If \(y=3^{x},\) then \(x=\sqrt[3]{y}\) b. \(\frac{\log _{b} x}{\log _{b} y}=\log _{b} x-\log _{b} y\) c. \(\log _{5} 4^{6}=4 \log _{5} 6\) d. \(2=10^{\log _{10} 2}\) e. \(2=\ln 2^{e}\) f. If \(f(x)=x^{2}+1,\) then .\(f^{-1}(x)=\frac{1}{x^{2}+1}\). g. If \(f(x)=1 / x,\) then \(f^{-1}(x)=1 / x\).

Inverse of composite functions a. Let \(g(x)=2 x+3\) and \(h(x)=x^{3} .\) Consider the composite function \(f(x)=g(h(x)) .\) Find \(f^{-1}\) directly and then express it in terms of \(g^{-1}\) and \(h^{-1}\). b. Let \(g(x)=x^{2}+1\) and \(h(x)=\sqrt{x}\). Consider the composite function \(f(x)=g(h(x)) .\) Find \(f^{-1}\) directly and then express it in terms of \(g^{-1}\) and \(h^{-1}\) c. Explain why if \(g\) and \(h\) are one-to-one, the inverse of \(f(x)=g(h(x))\) exists.

Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. $$ O \circ E $$
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