Problem 44 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph is a horizontal line at \( y = -1 \) for \( x < -1 \); a diagonal line from \((-1, -1)\) to \((1, 1)\); and a horizontal line at \( y = 1 \) for \( x > 1 \).

Step 1: Understand the Function

The given function is a piecewise function with three different expressions based on the value of \( x \):1. \( f(x) = -1 \) for \( x < -1 \).2. \( f(x) = x \) for \( -1 \leq x \leq 1 \).3. \( f(x) = 1 \) for \( x > 1 \). Each piece is valid only over its specified interval.

Step 2: Identify Intervals and Their Graphs

For each interval:- For \( x < -1 \), the function is constant at \( -1 \), which means it is a horizontal line at \( y = -1 \).- For \( -1 \leq x \leq 1 \), the function is \( f(x) = x \), which is a linear function represented by a diagonal line passing through \((-1, -1)\) and \((1, 1)\).- For \( x > 1 \), the function is constant at \( 1 \), which means it is a horizontal line at \( y = 1 \).

Step 3: Plot the Graph

Start by plotting the constant values:- Draw a horizontal line starting from \( x = -1 \) to \( x = -fty \) on the line \( y = -1 \).- Then, draw a diagonal line from \((-1, -1)\) to \((1, 1)\) representing \( f(x) = x \).- Finally, plot the horizontal line from \( x = 1 \) to \( x = fty \) on \( y = 1 \).Remember to place solid dots at any closed interval endpoints and open circles at open intervals.

Step 4: Graphical Description

The graph consists of three line segments:- A horizontal line on \( y = -1 \) extending leftward from \( x = -1 \) with an open circle.- A diagonal line connecting the points \((-1, -1)\) and \((1, 1)\), with solid dots at both these points.- A horizontal line on \( y = 1 \) extending rightward from \( x = 1 \) with an open circle.

Key Concepts

Graphing Functions

Graphing functions, especially piecewise functions, can seem daunting, but breaking it down into smaller steps simplifies the process. A piecewise function, like the one given, is defined by different expressions over various intervals of the domain. To graph it, follow these steps:

• Identify the different pieces of the function. For this piecewise function, you have three scenarios based on the value of \(x\).
• For \(x < -1\), the function is constant at \(f(x) = -1\), which means a horizontal line at \(y = -1\).
• In the interval \(-1 \leq x \leq 1\), the function is linear, \(f(x) = x\), represented by a diagonal line that passes through the points \((-1, -1)\) and \((1, 1)\).
• For \(x > 1\), the function is again constant at \(f(x) = 1\), so draw a horizontal line at \(y = 1\).

For each interval, observe the endpoints carefully. Closed intervals include endpoints as actual points on the graph, represented by filled dots, while open intervals exclude the endpoint, represented by open circles.

Continuous Functions

Continuous functions are graphs you can trace without lifting your pencil. They have no breaks, gaps, or jumps, making them very predictable. However, piecewise functions may not always be continuous depending on how their pieces connect.
In our exercise, the function is defined by three distinct pieces:

• The first segment (\(f(x) = -1\) for \(x < -1\)) ends with an open circle at \(x = -1\), indicating a discontinuity at this point.
• The segment \(-1 \leq x \leq 1\) is continuous as a straight line passes through both endpoints.
• For \(x > 1\), the horizontal line on \(y = 1\) starts with an open circle at \(x = 1\), creating another discontinuity between this segment and the previous one.

Thus, while individual segments of a piecewise function can be continuous, the function as a whole may not be. Continuity depends on how well the transitions between different pieces are connected.

Interval Notation

Interval notation is a mathematical shorthand for expressing subsets of real numbers, specifically indicating where certain sub-domains of a function are valid.
Let's breakdown the piecewise function:

• \(( -\infty, -1)\) signifies the open interval where \(f(x) = -1\). The open parenthesis \(()\) means \(-1\) is not included.
• \([-1, 1]\) indicates a closed interval where \(f(x) = x\). The use of square brackets \([]\) tells us that \(-1\) and \(1\) are included.
• \((1, \infty)\) tells us \(f(x) = 1\) beyond \(x = 1\), again using an open parenthesis to exclude \(1\) itself.

Using interval notation helps in simplifying the understanding of complex functions by clearly defining the start and stop of each segment. This way, you can easily pick which expression applies to which part of the function.