Problem 40 - College Algebra

Question

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. $$ f(x)=\left\\{\begin{array}{l}{x+1 \text { if } x<0} \\ {x-1 \text { if } x>0}\end{array}\right. $$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, 0) \cup (0, \infty)\).

Step 1: Identify the Conditions

The function is defined piecewise, meaning it has different expressions for different parts of the domain. The conditions in the problem are \(x < 0\) and \(x > 0\). There is no information given for \(x = 0\), indicating the function is not defined at this point.

Step 2: Write the Expressions

The function expressions are: \(f(x) = x + 1\) when \(x < 0\) and \(f(x) = x - 1\) when \(x > 0\). These are the lines to be drawn on the graph within their respective domains.

Step 3: Sketch the Graph for \(x < 0\)

For \(x < 0\), use the expression \(f(x) = x + 1\). This is a straight line with a slope of 1 and a y-intercept of 1. Draw this line starting from just above the point \(x = 0\) and continuing to the left.

Step 4: Sketch the Graph for \(x > 0\)

For \(x > 0\), use the expression \(f(x) = x - 1\). This line also has a slope of 1, but with a y-intercept of -1. Draw this line starting just to the right of the point \(x = 0\) and extending to the right.

Step 5: Write the Domain

The function is defined for all \(x\) except at \(x = 0\). Therefore, the domain of the function in interval notation is \((-\infty, 0) \cup (0, \infty)\).

Key Concepts

Domain in Interval Notation

Understanding the domain of a piecewise function is crucial in order to know where the function is defined. For piecewise functions, each piece can have its own domain, based on the conditions provided in the exercise.
In this example, the domain excludes the value at which the function is undefined, which is at 0. Thus, the function's domain excludes 0. To express this in interval notation, we write it as

• (-∞, 0) for values less than 0.
• (0, ∞) for values greater than 0.

This notation effectively states that any x-value except 0 is part of the domain, capturing continuous segments from negative to positive infinity.

Graphing Piecewise Functions

Graphing a piecewise function involves plotting each piece on its respective interval. For the exercise, there are two linear expressions:

• For values of x less than 0, we use the equation \( f(x) = x + 1 \). This represents a line with a slope of 1, crossing the y-axis at 1. When graphing, start just before 0 and draw leftwards.
• For values of x greater than 0, we apply the equation \( f(x) = x - 1 \). Similarly, it has a slope of 1 but intercepts the y-axis at -1. Begin slightly after 0 and extend the line to the right.

When examining the graph, notice that there is a gap at \( x = 0 \) because the function is not defined at this point. This is important as it visually represents the discontinuity in the function's domain.

Function Expressions

Piecewise functions are defined using different expressions on different intervals, often to describe behaviors that change based on the input. Every expression in a piecewise function corresponds to a specific condition of x.

• In the given exercise, we have two expressions: \( f(x) = x + 1 \) for \( x < 0 \), and \( f(x) = x - 1 \) for \( x > 0 \).
• These expressions are linear. Identifying them helps in knowing the rules that dictate the function's behavior over its domain.

When writing these expressions, it's essential to specify the conditions accurately. This ensures that when you graph the function, each part corresponds correctly to the interval designed for it. Understanding these expressions aids not only in sketching but also in interpreting the function's properties like slope and intercepts, important for comprehensive mathematical analysis.