Intervals Where G(x) = Cube Root Of (x-5) Is Negative
In the realm of mathematical functions, understanding the behavior of a function across different intervals is crucial. This article delves into the function g(x) = \sqrt[3]{x-5}, specifically focusing on identifying the interval where the function yields negative values. This exploration involves analyzing the properties of cube root functions and their relationship to the input values. By meticulously examining the function's structure, we can pinpoint the range of x-values for which g(x) is less than zero. This skill is fundamental in various mathematical applications, including calculus, real analysis, and problem-solving in general.
To effectively determine the interval where g(x) is negative, a solid understanding of the cube root function is essential. Unlike square roots, cube roots can accept negative numbers as input and produce real number outputs. This is because a negative number multiplied by itself three times results in a negative number. For example, the cube root of -8 is -2 because (-2) * (-2) * (-2) = -8. This property distinguishes cube roots from square roots, which yield imaginary numbers when applied to negative inputs. Understanding this core characteristic of cube roots is crucial for analyzing the behavior of g(x) = \sqrt[3]{x-5}.
The cube root function, denoted as \sqrt[3]x}, is the inverse operation of cubing a number. It essentially asks the question.
The behavior of the cube root function around zero is particularly important. For positive inputs, the cube root is positive, and for negative inputs, the cube root is negative. The cube root of zero is zero. This transition from negative to positive values as the input changes sign is a key characteristic that determines the intervals where functions involving cube roots are positive or negative. In the context of g(x) = \sqrt[3]{x-5}, this means we need to analyze when the expression inside the cube root, (x-5), is negative, zero, or positive to determine the sign of the function. This understanding forms the basis for solving the problem of finding the interval where g(x) is negative.
Given the function g(x) = \sqrt[3]{x-5}, our goal is to identify the interval where g(x) is negative. To achieve this, we need to analyze the expression inside the cube root, which is (x-5). The cube root function, as discussed earlier, returns a negative value when its input is negative. Therefore, g(x) will be negative when (x-5) is negative. This insight is crucial for setting up the inequality that will help us determine the desired interval.
To find the interval where (x-5) is negative, we set up the inequality x - 5 < 0. Solving this inequality involves isolating x by adding 5 to both sides of the inequality. This gives us x < 5. This inequality indicates that for any x-value less than 5, the expression (x-5) will be negative. Consequently, the cube root of (x-5) will also be negative. This is because the cube root of a negative number is negative. Therefore, the interval where g(x) is negative is directly related to the solution of this inequality.
The solution x < 5 represents all real numbers less than 5. In interval notation, this is expressed as (-∞, 5). This interval includes all numbers from negative infinity up to, but not including, 5. At x = 5, the expression (x-5) becomes zero, and the cube root of zero is zero, so g(x) is zero at x = 5. For x values greater than 5, the expression (x-5) becomes positive, and the cube root of a positive number is positive, making g(x) positive. Therefore, the interval (-∞, 5) is the precise interval where g(x) is negative. This analysis demonstrates how understanding the properties of the cube root function and solving simple inequalities can lead to determining the intervals where a function is negative.
Based on the analysis above, we've established that g(x) = \sqrt[3]{x-5} is negative when x - 5 < 0. Solving this inequality, we get x < 5. This means the function is negative for all x-values less than 5. In interval notation, this is represented as (-∞, 5). This interval includes all real numbers from negative infinity up to, but not including, 5.
To further illustrate this, let's consider a few examples. If x = 4, then g(4) = \sqrt[3]{4-5} = \sqrt[3]{-1} = -1, which is negative. If x = 0, then g(0) = \sqrt[3]{0-5} = \sqrt[3]{-5}, which is also negative. However, if x = 6, then g(6) = \sqrt[3]{6-5} = \sqrt[3]{1} = 1, which is positive. These examples provide concrete evidence that the function g(x) is indeed negative for x-values less than 5 and positive for x-values greater than 5.
The interval (-∞, 5) accurately captures the range of x-values for which g(x) is negative. The parenthesis around 5 indicates that 5 is not included in the interval, which is consistent with our earlier observation that g(5) = 0. This precise determination of the negative interval is crucial for various applications, such as graphing the function, solving inequalities involving g(x), and understanding the function's behavior in different contexts. Therefore, the correct answer is the interval (-∞, 5), which clearly defines the region where the function g(x) yields negative values.
In conclusion, by carefully analyzing the function g(x) = \sqrt[3]{x-5} and leveraging our understanding of cube root functions, we have successfully identified the interval where the function is negative. The key was to recognize that the cube root function returns a negative value when its input is negative. By setting up and solving the inequality x - 5 < 0, we determined that g(x) is negative for all x-values less than 5. This is represented by the interval (-∞, 5).
This exercise highlights the importance of understanding the properties of fundamental functions, such as the cube root function, and how these properties influence the behavior of more complex functions. The ability to determine intervals where a function is positive, negative, or zero is a fundamental skill in mathematics, with applications in calculus, analysis, and problem-solving in general. The process of analyzing the function, setting up the appropriate inequality, and solving for the variable is a valuable technique that can be applied to a wide range of mathematical problems.
Furthermore, this exploration emphasizes the significance of interval notation as a precise way to represent a set of real numbers. The interval (-∞, 5) clearly communicates that the set includes all real numbers less than 5, without ambiguity. The parenthesis around 5 indicates that 5 is not included in the set, which is crucial for accurately describing the behavior of the function at that point. By mastering these concepts and techniques, students can develop a deeper understanding of mathematical functions and their applications.
The final answer is B. (-∞, 5).
