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Chapter 1: Problem 138

Define a piecewise function on the intervals \((-\infty, 2],(2,5)\) and \([5, \infty)\) that does not "jump" at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.

Short Answer

Expert verified
The final piecewise function which meets the given requirements is: \( f(x) = \begin{cases} 1 & \text{if } x \leq 2 \\ x - 1 & \text{if } 2 < x < 5 \\ 9 - x & \text{if } x \geq 5 \end{cases} \)

Step by step solution

01

Define the constant function

In the interval \((-\infty, 2]\), let's define a constant function: \(f(x)=c\), where 'c' can be any real number. For simplicity, let's choose \(c=1\). Thus, the function \(f(x)=1\) is defined for every \(x\leq2\).
02

Define the increasing function

For the second part of the overall function in the interval \((2,5)\), let's define an increasing linear function that connects with our constant function at the point \(x=2\). A function like \(f(x)=x-1\) would work, as it increases as 'x' increases and \(f(2)=1\), thus our function does not 'jump' at \(x=2\).
03

Define the decreasing function

In the third interval \([5,\infty)\), we will define a decreasing function. A good example of such a function is a line with a negative slope. To ensure that the function does not jump at \(x=5\), we should take a linear function that equals to 4 at \(x=5\). Hence, \(f(x)=9-x\) will work for this situation.

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Key Concepts

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

Constant Function
A constant function is one that retains the same value, no matter what input is given. In simpler terms, it remains flat on the graph, like a completely horizontal line. This makes it a pretty straightforward concept to grasp. In the realm of piecewise functions, a constant function will maintain this flat line across a given interval.
  • For example, if we define a constant function as \( f(x) = c \) where \( c \) is a real number, it means for any value that \( x \) takes on, \( f(x) \) will always equal \( c \).
  • In our specific exercise example, we set \( f(x) = 1 \) for all \( x \leq 2 \), effectively keeping the function flat at the height '1'.
An important feature of the constant function within a piecewise segment is that it helps prevent 'jumps.' Since its value does not change, it smoothly transitions into adjoining function segments. This property is especially valuable in ensuring that piecewise functions are continuous at points of transition.
Increasing Function
An increasing function describes how the function's value climbs as the input value grows. Graphically, this appears as an upward slope from left to right. This linear growth pattern is beneficial when defining a part of a piecewise function.
  • For instance, a simple linear increasing function would look like \( f(x) = mx + b \), where 'm' is a positive slope that determines how steeply this line inclines.
  • In the given problem, we choose \( f(x) = x - 1 \) between \( (2,5) \).
This choice ensures continuity, meaning the function neatly picks up where the constant function leaves off, without skipping or causing a sudden jump at \( x = 2 \). The function begins at \( f(2) = 1 \) and smoothly increases, which sets up a seamless transition to any following sections of the piecewise function.
Decreasing Function
A decreasing function characterizes how the value of the function drops as the input increases. This is the opposite of an increasing function, and on a graph, it forms a downward slope as you move from left to right.
  • In mathematical terms, a decreasing function might be represented by \( f(x) = mx + b \), where 'm' is a negative slope.
  • The specific exercise uses \( f(x) = 9 - x \) for the interval \( [5, \infty) \), which begins at \( f(5) = 4 \) and decreases as \( x \) grows larger.
It is critical to ensure that the decreasing function meshes appropriately with the ending value of the preceding piece of the function. This avoids any abrupt changes or discontinuity at the point \( x = 5 \). This way, the series of piecewise functions look smooth and continuous, ensuring a logical progression across different input intervals.

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