Problem 43

Question

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{-1} & {\text { if } x<-1} \\ {1} & {\text { if }-1 \leq x \leq 1} \\ {-1} & {\text { if } x>1}\end{array}\right. $$

Step-by-Step Solution

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Answer

A graph consisting of horizontal lines: $ y = -1 $ for $ x < -1 $ and $ x > 1 $, $ y = 1 $ for $ -1 \leq x \leq 1 $.

1Step 1: Understand the Piecewise Function

First, let's analyze the function piecewise. We have three distinct conditions: - For $ x < -1 $, the function $ f(x) = -1 $.- For $ -1 \leq x \leq 1 $, the function $ f(x) = 1 $.- For $ x > 1 $, the function $ f(x) = -1 $. This means the function changes behavior in these intervals.

2Step 2: Determine the Values at Interval Boundaries

Let's find the function's value at the boundaries of the intervals:- As $ x $ approaches $-1$ from the left, $ f(x) = -1 $. At $ x = -1 $, $ f(x) = 1 $.- At $ x = 1 $, $ f(x) = 1 $. As $ x $ moves past 1, $ f(x) = -1 $.

3Step 3: Sketch Each Interval's Part

To sketch the graph, draw horizontal lines for each interval:- Draw a horizontal line at $ y = -1 $ for $ x < -1 $. This line is open at $ x = -1 $ because the value changes to 1.- Draw a horizontal line at $ y = 1 $ from $ x = -1 $ to $ x = 1 $, including the endpoints.- Draw a horizontal line at $ y = -1 $ for $ x > 1 $. This line is open at $ x = 1 $ because the value changes from 1.

4Step 4: Mark Open and Closed Circles

Mark circles to indicate endpoints:- At $ x = -1 $, place an open circle at $ y = -1 $ and a closed circle at $ y = 1 $, indicating the change from $-1$ to $1$.- At $ x = 1 $, place a closed circle at $ y = 1 $ and an open circle at $ y = -1 $, reflecting the switch from $1$ back to $-1$.

Key Concepts

Intervals in Piecewise FunctionsGraphing Piecewise FunctionsEndpoint Notation in Graphs

Intervals in Piecewise Functions

When dealing with piecewise functions, the term "interval" refers to the specific range of input values (or x-values) over which a particular expression or rule applies. In a piecewise function, the domain is divided into several intervals. Each interval is associated with a distinct function definition.
For the function given:

For the interval where $x < -1$, $f(x)$ is defined as $-1$.
In the interval $-1 \leq x \leq 1$, $f(x)$ takes the value $1$.
The interval $x > 1$ again sets $f(x)$ to $-1$.

Each of these intervals operates independently. Understanding these is crucial for evaluating the function correctly as the interval determines which piece of the function to use. Paying attention to the symbols "$<$" and "$\leq$" is important because they dictate whether endpoints are included or not.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting each segment as per its defined interval on a coordinate plane. Start by focusing on each piece individually. For each function segment, draw a separate part of the graph corresponding to its specific interval and expression.
For our example:

For $x < -1$, plot a horizontal line at $y = -1$. This line stops short at $x = -1$ because of the inequality.
Between $-1$ and $1$, draw another horizontal line at $y = 1$. This includes the endpoints $x = -1$ and $x = 1$, covering values in the interval.
Beyond $x = 1$, revert to drawing a line at $y = -1$. This portion starts just after $x = 1$, corresponding to the last part of the function.

By taking each interval step by step, you build the full picture of the piecewise function visually. Ensure to account for any starts or stops in the line when the interval changes.

Endpoint Notation in Graphs

Endpoints in piecewise function graphs are where each segment starts or stops at the boundary of an interval. Endpoint notation is critical when sketching these graphs as it clarifies inclusion or exclusion of boundary points. This is where open and closed circles come into play.
Here's the guide for endpoints:

An open circle (◯) represents an endpoint that is not included in the interval. This happens if it's a "<" or ">" sign.
A closed circle (•) signifies an included endpoint, meaning that point is part of the interval and will have a "≤" or "≥" used.

For our graph:

At $x = -1$, place an open circle for the interval $x < -1$ segment and a closed circle for the $-1 \leq x \leq 1$ part, marking the transition from $-1$ to $1$.
Similarly, at $x = 1$, a closed circle will be on $y = 1$ as $x$ is included in that interval, and an open circle on $y = -1$ as it marks the transition.

Proper endpoint notation ensures that the graph accurately represents the piecewise function and avoids confusion about which values the function actually reaches.

Problem 43

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