Problem 53 - College Algebra

Question

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

Step-by-Step Solution

Answer

The domain of the piecewise function is \((-\infty, 1) \cup (1, \infty)\).

Step 1: Understanding the Function

We have a piecewise function defined as: \[ f(x)= \begin{cases} x^{2}-2 & \text{if } x<1 \ -x^{2}+2 & \text{if } x>1 \end{cases} \] The function has two expressions, each valid for different ranges of \(x\).

Step 2: Identify Domain for Each Part

The first part of the function is \( x^2 - 2 \), and it is defined for \( x < 1 \). The second part is \( -x^2 + 2 \), and it is defined for \( x > 1 \). In this piecewise function, there is no definition for \( x = 1 \).

Step 3: Combine Intervals

The domain of \( f(x) \) is the union of the domains of each part of the piecewise function. For the given piecewise function: - The interval for \( x^2 - 2 \) is \( (-\infty, 1) \). - The interval for \( -x^2 + 2 \) is \( (1, \infty) \).

Key Concepts

Understanding the Domain of a Function

The domain of a function includes all the possible values of the independent variable, often represented by \( x \), for which the function is defined. Simply put, it's the set of all input values that will not cause issues like division by zero or taking the square root of a negative number.

In our piecewise function example, the domain is constructed by examining each piece of the function separately. We have:

• \( x^2 - 2 \) valid for \( x < 1 \)
• \( -x^2 + 2 \) valid for \( x > 1 \)

The challenge here is understanding that there is no piece of the function that covers \( x = 1 \), as neither condition includes this point. Thus, it's excluded from the domain.

To express the domain of this piecewise function in mathematical terms, we combine the intervals from each piece, excluding \( x = 1 \). This gives us the domain as the union of \((-\infty, 1)\) and \((1, \infty)\), covering all real numbers except \( x = 1 \). Understanding domains like this is crucial in piecewise functions and ensures the function makes logical sense across its range.

The Role of Interval Notation

Interval notation is a concise way of describing ranges of values, often used to express domains and ranges of functions. It uses brackets and parentheses to show the range boundaries:

• Round brackets \((a, b)\) indicate that the endpoints \(a\) and \(b\) are not included in the interval.
• Square brackets \([a, b] \) mean that endpoints \(a\) and \(b\) are included.
• Combining them, such as \((a, b] \) or \([a, b)\), indicates inclusion of one endpoint and exclusion of the other.

In our piecewise function, because it is not defined at \( x = 1 \), we use parentheses, indicating this value is not included. Therefore, the domain in interval notation is written as \((-\infty, 1) \cup (1, \infty)\). This conveys that the function accepts every real number except \( x = 1 \).

Interval notation is not limited to expressing just domains. It also neatly demonstrates other ranges and is a versatile tool in set theory and calculus. Understanding this notation allows students to communicate mathematical ideas clearly and efficiently.

Algebraic Functions in Piecewise Contexts

Algebraic functions are built from basic operations such as addition, subtraction, multiplication, division, and root extraction. In piecewise functions, different algebraic rules are applied in separate sections of the domain.

In our exercise, we see this with:

• \( x^2 - 2 \) when \( x < 1 \), representing a simple parabola shifted downward by 2 units.
• \( -x^2 + 2 \) for \( x > 1 \), depicting an inverted parabola moved upward by 2 units.

Piecewise functions allow for different rules governing different segments of their domains. They are useful when situations do not align with one single mathematical expression. For example, real-world models like a taxi fare system could be piecewise when using different rates for varying distances.

Understanding these functions requires recognizing that each piece's domain dictates when it applies. Separately analyzing each segment, investigating how they interact at endpoints is key. For instance, realizing \( x = 1 \) is exclusive means identifying the piecewise boundaries are crucial for a precise depiction of the function's behavior. These algebraic functions in a piecewise arrangement demonstrate how a single function expresses varied mathematical principles across distinct intervals.