Understanding The Function F(x) = √(-x) Domain And Range Analysis

In this comprehensive analysis, we will delve into the intricacies of the function f(x) = √(-x), exploring its domain, range, and how it relates to other functions. Understanding these aspects is crucial for grasping the behavior of this function and its applications in various mathematical contexts. We will examine the function in detail, comparing it to related functions and ultimately determining which statement about it is true. This exploration will not only solidify your understanding of this particular function but also enhance your general knowledge of function analysis.

Exploring the Domain of f(x) = √(-x)

When analyzing functions, a crucial first step is to determine the domain. The domain of a function is the set of all possible input values (x-values) for which the function produces a real output. For the function f(x) = √(-x), we encounter a specific constraint due to the presence of the square root. Recall that the square root of a negative number is not a real number; it results in an imaginary number. Therefore, the expression inside the square root, -x, must be greater than or equal to zero.

Mathematically, this condition is expressed as -x ≥ 0. To solve this inequality for x, we can multiply both sides by -1. Remember that when multiplying or dividing an inequality by a negative number, we must reverse the inequality sign. This gives us x ≤ 0. Consequently, the domain of f(x) = √(-x) consists of all real numbers less than or equal to zero. This means that we can input any non-positive number into the function and obtain a real output, while positive numbers would lead to imaginary results. Understanding this restriction is fundamental to accurately interpreting the function's behavior and its graph.

In simpler terms, the function f(x) = √(-x) is defined only for x-values that are zero or negative. If we were to try plugging in a positive value, such as x = 1, we would get f(1) = √(-1), which is not a real number. On the other hand, if we input x = -4, we get f(-4) = √(-(-4)) = √(4) = 2, a real number. This illustrates the domain restriction in action. The domain plays a significant role in how we visualize the function's graph, as it tells us the region of the x-axis where the graph exists. This careful consideration of the domain is a cornerstone of function analysis, ensuring we work with valid inputs and outputs.

Analyzing the Range of f(x) = √(-x)

Having established the domain, the next key aspect to analyze is the range of the function f(x) = √(-x). The range represents the set of all possible output values (y-values) that the function can produce. Given that the function involves a square root, we know that the output will always be non-negative. This is because the principal square root of any non-negative number is, by definition, non-negative.

Consider the expression √(-x). Since -x must be greater than or equal to zero (as we determined in the domain analysis), the square root of -x will also be greater than or equal to zero. Therefore, the function f(x) = √(-x) will only produce non-negative output values. The smallest possible output value is 0, which occurs when x = 0 (f(0) = √(0) = 0). As x becomes more negative, the value of -x becomes more positive, and the square root of -x increases. This means the function can take on any non-negative value.

In conclusion, the range of the function f(x) = √(-x) is all non-negative real numbers, often expressed as [0, ∞). This implies that the graph of the function will exist only in the upper half of the coordinate plane (where y ≥ 0). The range provides crucial information about the function's behavior and its graphical representation. Understanding the range helps us anticipate the possible output values and how the function transforms input values. It's an essential component in the complete characterization of a function, complementing the domain in providing a full picture of its behavior.

Comparing f(x) = √(-x) with f(x) = -√(-x)

Now, let's compare the function f(x) = √(-x) with the function g(x) = -√(-x). By carefully examining their properties, we can determine whether they share the same domain, range, or other characteristics. This comparative analysis is crucial for understanding how seemingly small changes to a function's formula can impact its behavior.

Domain Comparison

As we established earlier, the domain of f(x) = √(-x) is all x ≤ 0. For the function g(x) = -√(-x), the expression inside the square root, -x, must also be greater than or equal to zero. This is because we cannot take the square root of a negative number and obtain a real result. Therefore, the condition -x ≥ 0 must hold true, which implies x ≤ 0. Consequently, the domain of g(x) = -√(-x) is also all x ≤ 0.

Comparing the domains, we find that both f(x) = √(-x) and g(x) = -√(-x) have the same domain: all real numbers less than or equal to zero. This means that both functions are defined for the same set of input values. This shared domain is a fundamental similarity between the two functions.

Range Comparison

Now, let's consider the range of g(x) = -√(-x). We know that the square root part, √(-x), will always be non-negative (greater than or equal to zero). However, the negative sign in front of the square root, -√(-x), means that the output of g(x) will always be non-positive (less than or equal to zero). The smallest possible output value occurs when x = 0, where g(0) = -√(0) = 0. As x becomes more negative, the value of -x becomes more positive, and the square root of -x increases. However, due to the negative sign, the entire expression becomes more negative. Therefore, the range of g(x) = -√(-x) is all non-positive real numbers, which can be expressed as (-∞, 0].

Comparing the ranges, we see a significant difference. The range of f(x) = √(-x) is [0, ∞), while the range of g(x) = -√(-x) is (-∞, 0]. This means that f(x) can only produce non-negative outputs, while g(x) can only produce non-positive outputs. This difference in range is a key distinction between the two functions.

Key Differences and Similarities

In summary, f(x) = √(-x) and g(x) = -√(-x) share the same domain (x ≤ 0) but have different ranges. f(x) has a range of [0, ∞), while g(x) has a range of (-∞, 0]. This implies that g(x) is essentially a reflection of f(x) across the x-axis. The negative sign in g(x) flips the output values, changing the range from non-negative to non-positive.

Determining the True Statement

Based on our analysis, we can now evaluate the given statements about the function f(x) = √(-x). Let's revisit the original question:

Which statement is true about the function f(x) = √(-x)?

A. It has the same domain as the function f(x) = -√(-x). B. It has the same range as the function f(x) = -√(-x). C. It has the same domain as the function.

We have already established that f(x) = √(-x) and g(x) = -√(-x) share the same domain (x ≤ 0) but have different ranges. Therefore, statement A is true, and statement B is false. Statement C is incomplete and lacks a function for comparison, making it difficult to evaluate. Therefore, based on our analysis, the true statement is:

A. It has the same domain as the function f(x) = -√(-x).

This conclusion highlights the importance of carefully analyzing the domain and range of functions when comparing them. Understanding these properties allows us to accurately describe the behavior of functions and their relationships to each other. This analysis also underscores the impact of seemingly minor changes, such as a negative sign, on a function's characteristics.

Conclusion

In this comprehensive analysis, we have explored the function f(x) = √(-x) in detail. We determined its domain to be all x ≤ 0 and its range to be [0, ∞). We then compared it with the function g(x) = -√(-x), highlighting that while they share the same domain, their ranges differ significantly. This led us to conclude that statement A, which states that f(x) = √(-x) has the same domain as f(x) = -√(-x), is the true statement.

This exercise demonstrates the importance of thoroughly analyzing the properties of functions, including their domains and ranges, to understand their behavior and relationships. By carefully examining these characteristics, we can make accurate comparisons and draw meaningful conclusions. This approach is fundamental to advanced mathematical concepts and problem-solving, providing a solid foundation for further exploration in mathematics.

The understanding of domains and ranges, as demonstrated in this analysis, is crucial not only for theoretical mathematics but also for practical applications. In various fields, such as physics, engineering, and computer science, functions are used to model real-world phenomena. Knowing the domain and range of these functions ensures that the models are accurate and meaningful. This underscores the value of a strong grasp of function analysis in a wide range of disciplines.