Domain Of Piecewise Function F(x) A Comprehensive Analysis

In mathematics, piecewise functions are defined by multiple sub-functions, each applying to a specific interval of the domain. Understanding these functions requires a careful examination of their definitions and the intervals over which they apply. Let's consider the given piecewise function:

$f(x)=\left\{\begin{array}{cl}-x-2, & x < -2 \\-x^2, & -2 < x < 0 \\x, & x \geq 0\end{array}\right.$

To fully grasp this function, we need to break it down and analyze each piece. This article will delve into the domain of this piecewise function, providing a comprehensive explanation and addressing any potential questions.

Analyzing the Piecewise Function

This function, denoted as f(x), is composed of three distinct sub-functions, each with its own specific domain:

1. For $x < -2$, the function is defined as $f(x) = -x - 2$. This is a linear function with a slope of -1 and a y-intercept of -2. It is valid for all values of x less than -2.
2. For $-2 < x < 0$, the function is defined as $f(x) = -x^2$. This is a quadratic function, specifically a downward-opening parabola. It is valid for all values of x between -2 and 0, excluding -2 and 0.
3. For $x \geq 0$, the function is defined as $f(x) = x$. This is a simple linear function with a slope of 1, passing through the origin. It is valid for all values of x greater than or equal to 0.

Each of these sub-functions contributes to the overall behavior of f(x) across its domain. Understanding how these pieces fit together is crucial for determining the function's domain, range, and other properties.

What is the Domain of a Function?

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's the collection of all x-values that you can plug into the function and get a valid output (y-value).

For example, consider the function $f(x) = \frac{1}{x}$. The domain of this function is all real numbers except for 0, because division by zero is undefined. Similarly, for the function $g(x) = \sqrt{x}$, the domain is all non-negative real numbers, since the square root of a negative number is not a real number.

When dealing with piecewise functions, determining the domain involves considering the domains of each individual piece and combining them. We need to ensure that there are no gaps or overlaps in the domain, and that the function is defined for all values within its specified intervals.

Key Considerations for Determining the Domain

When identifying the domain of any function, especially piecewise functions, there are a few critical aspects to consider:

• Division by Zero: Rational functions (functions involving fractions) are undefined when the denominator is zero. Therefore, any x-value that makes the denominator zero must be excluded from the domain.
• Square Roots of Negative Numbers: In the realm of real numbers, the square root of a negative number is undefined. Consequently, any x-value that results in a negative number under a square root must be excluded from the domain.
• Interval Boundaries: For piecewise functions, it's essential to carefully examine the boundaries of each interval. The function's definition might change at these boundaries, and it's crucial to ensure that the function is defined consistently.
• Overlapping Intervals: The intervals in a piecewise function should not overlap. If they do, the function would have multiple definitions for the same x-value, which is not allowed.
• Gaps in the Domain: The domain should be continuous, with no gaps or holes, unless explicitly defined. If there are any values of x for which the function is undefined, these values must be excluded from the domain.

By carefully considering these factors, we can accurately determine the domain of a function, whether it's a simple algebraic function or a more complex piecewise function.

Determining the Domain of f(x)

To find the domain of the given piecewise function, we need to analyze the intervals for which each piece is defined:

1. The first piece, $f(x) = -x - 2$, is defined for $x < -2$. This means all real numbers less than -2 are included in the domain.
2. The second piece, $f(x) = -x^2$, is defined for $-2 < x < 0$. This includes all real numbers between -2 and 0, but not -2 and 0 themselves.
3. The third piece, $f(x) = x$, is defined for $x \geq 0$. This includes all real numbers greater than or equal to 0.

Now, let's combine these intervals. The first interval includes all numbers less than -2, the second includes numbers between -2 and 0, and the third includes numbers greater than or equal to 0. Notice that there are no gaps or overlaps between these intervals. The function is defined for all real numbers less than -2, between -2 and 0, and greater than or equal to 0.

Therefore, the domain of the function f(x) is the set of all real numbers. We can express this in interval notation as $(-\infty, -2) \cup (-2, 0) \cup [0, \infty)$, which simplifies to $(-\infty, \infty)$.

Expressing the Domain

There are several ways to express the domain of a function:

• Set Notation: This notation uses set symbols to define the domain. For the function f(x), the domain in set notation is {x | x ∈ ℝ}, which reads as