Problem 52 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The domain is \((-\infty, -2) \cup [-2, \infty)\)."

Step 1: Understand the Piecewise Function

The given function \( f(x) \) has two different expressions based on the value of \( x \). These expressions are separated by the condition \( x = -2 \). Thus, \( f(x) = x + 1 \) when \( x < -2 \) and \( f(x) = -2x - 3 \) when \( x \geq -2 \).

Step 2: Determine the Domain Interval for Each Piece

1. For the expression \( x + 1 \), the function is defined for \( x < -2 \). This corresponds to the interval \(( -\infty, -2 )\).2. For the expression \( -2x - 3 \), the function is defined for \( x \geq -2 \). This corresponds to the interval \([-2, \infty)\).

Step 3: Combine Intervals to Define the Domain

Since the piecewise function covers all \( x \) values through its conditions, combine the two intervals \(( -\infty, -2 )\) and \([-2, \infty)\). Therefore, the function's domain in interval notation is \(( -\infty, -2 ) \cup [-2, \infty)\).

Key Concepts

Understanding Interval Notation

When discussing the domain of a function, one useful way to represent it is through interval notation. Interval notation provides a concise way of expressing the set of input values, or potential values of \( x \), for which a function is defined.
To get started, there are two types of intervals you should know about: open intervals and closed intervals. An open interval, such as \((a, b)\), includes all numbers between \(a\) and \(b\) but not \(a\) or \(b\) themselves. Conversely, a closed interval, like \([a, b]\), includes both endpoints \(a\) and \(b\) as well as all numbers in between.
There are also half-open intervals, which include one endpoint but not the other, represented as \([a, b)\) or \((a, b]\). With this exercise, the function spans different intervals that can be combined using the union symbol \(\cup\) to describe discontinuous domains. So, the complete domain of a piecewise function can be expressed as a union of multiple intervals, capturing the entire set of allowable input values.

Decoding Piecewise Functions

Piecewise functions are interesting mathematical constructs because their rules change based on different conditions of the input value \( x \). They're essentially a collection of separate function expressions, each with its own domain, working harmoniously to define a single function.
This type of function can look intimidating at first, but it is simply a means to express a situation where a function behaves differently over different intervals of \( x \). In the context of this task, the expression \( f(x) = x + 1 \) for \( x < -2 \) describes how the function behaves for all values less than \(-2\), while \( f(x) = -2x - 3 \) for \( x \geq -2 \) applies to values \(-2\) and greater.
Ultimately, combining these "pieces" brings together separate domains to form a continuous one for the entire function, which can then be described using interval notation. Remember, each piece of a piecewise function is valid only over its specified interval.

Exploring the Algebra Domain

Defining the domain of a piecewise function is an essential skill in algebra, focusing on identifying all possible inputs for which the function is defined, performing real number calculations without resulting in undefined behaviors.

• The domain is crucial because it sets the boundary for where the function operates without issues, such as division by zero or taking the square root of a negative number—situations that don't apply here but are important to consider in different contexts.
• For the given piecewise function, the domain includes all real numbers since both pieces cover different overlapping and non-overlapping intervals. This means there are no restrictions other than those explicitly stated in the piecewise conditions.

The domain in this specific case is written in interval notation as \((-\infty, -2) \cup [-2, \infty)\), denoting that \(-2\) acts as the boundary point where one expression stops being valid, and another starts.