Problem 33 - Calculus with Concepts in Calculus

Question

Find the domain of the function. $$ f(x)=\left\\{\begin{array}{l} 2 x \text { for }-4 \leq x \leq-1 \\ 3 \text { for } 0

Step-by-Step Solution

Answer

The domain is \([-4, -1] \cup (0, 6)\).

Step 1: Analyze each piece of the piecewise function

The given function is a piecewise function with two parts: 1. The first piece is given by \( 2x \) and is defined for \( -4 \leq x \leq -1 \).2. The second piece is a constant function \( 3 \) defined for \( 0 < x < 6 \).We need to consider the domain from the conditions of each piece.

Step 2: Determine the domain of the first piece

For the first piece \( 2x \), the domain is explicitly given as \( -4 \leq x \leq -1 \). This means this part of the function is defined for all \( x \) values from \(-4\) to \(-1\), including \(-4\) and \(-1\).

Step 3: Determine the domain of the second piece

The second function piece is constant with a value of 3, defined for \( x \) values in the interval \( 0 < x < 6 \). This means \( x \) can take any value strictly between 0 and 6, but neither 0 nor 6 are included.

Step 4: Combine the domains of both pieces

To find the domain of the entire piecewise function, combine the intervals from both pieces. The domain for the first piece is \([-4, -1]\) and for the second piece is \((0, 6)\). Thus, the overall domain for \( f(x) \) is the union of these intervals: \([-4, -1] \cup (0, 6)\).

Key Concepts

Piecewise Function

A piecewise function is a type of function that is defined by different expressions or rules over different intervals. Each "piece" of the function applies to a specific range of values of the variable, usually indicated by conditions on the variable.

In the context of our exercise, the function is defined by two distinct parts:

• For values of x between -4 and -1 (inclusive), the function is described as \( f(x) = 2x \).
• In the interval from 0 to 6 (exclusive), the function takes a constant value \( f(x) = 3 \).

Breaking the function into these pieces helps manage different behaviors or values the function takes over its domain.

Such functions are helpful for modeling situations where a relationship changes at certain key values, such as tax brackets or tiered pricing systems.

Interval Notation

Interval notation is a concise way of representing subsets of the real number line. It uses parentheses or brackets to show whether endpoints are included or excluded from the interval.

For instance, in our problem:

• The interval \([-4, -1]\) includes both endpoints -4 and -1. This is indicated by the square brackets.
• The interval \((0, 6)\) excludes both endpoints 0 and 6, as shown by the parentheses.

These notations efficiently communicate where each piece of the function is applicable within the domain.
Using interval notation helps to avoid confusion, providing a clear and universally understood description of the set of values that x can take.

Function Analysis

Function analysis involves examining the components, behavior, and characteristics of a function. It helps in understanding how a function behaves, where it is defined, and identifying any limitations in its domain.

In analyzing our piecewise function, we started by identifying the individual parts and their respective domains.

• The first part \( 2x \) operates over the interval \([-4, -1]\), meaning it accepts every real number between -4 and -1, inclusive.
• The second part \( f(x) = 3 \) covers the range \( (0, 6) \), a stretch where x must be greater than 0 and less than 6.

By examining each piece separately, we compiled these domains to determine the overall domain of the piecewise function, expressed as \([-4, -1] \cup (0, 6)\).
This step-by-step breakdown enables a comprehensive understanding of where and how the function operates, which is essential for solving equations, graphing, and real-world applications.