Problem 52 - College Algebra

Question

For the following exercises, write the domain for the piecewise function in interval notation. $$ f(x)=\left\\{\begin{array}{lll}{x+1} & {\text { if }} & {x<-2} \\ {-2 x-3} & {\text { if }} & {x \geq-2}\end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Piecewise Function

The piecewise function is defined by two different expressions based on the value of \(x\). For \(x < -2\), the expression is \(x + 1\). For \(x \geq -2\), the expression is \(-2x - 3\).

Step 2: Analyze the Domain for the First Expression

The condition for the first expression \(x + 1\) is \(x < -2\). Therefore, the domain for this part of the piecewise function is all \(x\) such that \(x\) is less than \(-2\), which we can represent as \((-\infty, -2)\).

Step 3: Analyze the Domain for the Second Expression

The condition for the second expression \(-2x - 3\) is \(x \geq -2\). Therefore, the domain for this part of the piecewise function is all \(x\) such that \(x\) is greater than or equal to \(-2\), which we can represent as \([-2, \infty)\).

Step 4: Combine the Domains

The overall domain of the piecewise function is the union of the domains of its parts. As one part covers \(x < -2\) and the other includes \(x \geq -2\), together they cover all real numbers. In interval notation, the complete domain of the piecewise function is \((-\infty, \infty)\).

Key Concepts

Domain

Understanding the domain of a function is essential for grasping its behavior. The domain of a function is the complete set of possible input values (usually denoted by the variable \( x \)) that satisfy the function. For piecewise functions, the domain can often be split into intervals defined by different expressions, each corresponding to a specific condition.In our exercise, the piecewise function consists of two expressions based on the value of \( x \):

• \( x + 1 \) for \( x < -2 \)
• \( -2x - 3 \) for \( x \geq -2 \)

The domain of this piecewise function is determined by combining the domains of these individual parts. For \( x + 1 \), the domain is \( x < -2 \), representing an interval of all numbers smaller than \(-2\). For \(-2x - 3\), the domain is \( x \geq -2 \), capturing all numbers starting from \(-2\) and larger. Together, these domains embrace all real numbers, from negative infinity to positive infinity.

Interval Notation

Interval notation is a convenient way to describe a range of numbers. It streamlines the expression of domains especially for piecewise functions by showing exactly which sections of the number line a function addresses.To read and write interval notation effectively, note:

• The interval \((a, b)\) represents all numbers greater than \(a\) and less than \(b\). It uses parentheses to show that \(a\) and \(b\) themselves are not included.
• The interval \([a, b]\) includes both \(a\) and \(b\), marked by square brackets.
• \((a, b]\) or \([a, b)\) can be used if only one endpoint is included.
• To indicate an interval extending indefinitely in a positive direction, use \(\infty\); for negatives, use \(-\infty\).

In our situation, the domain of the overall function spans all real numbers, expressed as \((-\infty, \infty)\) in interval notation. This notation concisely captures the function's ability to accept any real number as input.

Piecewise Expression

A piecewise function is defined by multiple sub-functions, each applying to a specific part of the domain. These functions are remarkably useful for modeling situations where a relationship between variables changes based on different conditions or intervals. Consider our example function \( f(x) \):

• For \( x < -2 \): the expression is \( x + 1 \). This applies strictly for values of \( x \) that are less than \(-2\).
• For \( x \geq -2 \): it changes to \(-2x - 3 \). This applies for any \( x \) greater than or equal to \(-2\).

Each piece of the piecewise function is valid in its own interval and seamlessly continues to the next as \( x \) transitions through the conditions. The conditions like \( x < -2 \) or \( x \geq -2 \) clearly define where each expression is applicable. Identifying and graphing these pieces provides insight into how the function behaves across its entire domain, giving a complete view of how sub-conditions interact within the function's framework.