Analyzing The Properties Of Function G(x) A Comprehensive Guide

Which statements are true about the function g(x)?

Before diving into the properties of the function, let's define function g(x) clearly. This function is defined piecewise, meaning it has different rules for different parts of its domain. For x values less than 0, the function is defined as g(x) = (1/2)^x + 3. For x values greater than or equal to 0, the function is defined as g(x) = -x^2 + 2. The transition point is at x = 0, where the function's behavior changes. Understanding this piecewise definition is crucial for analyzing the function's properties, such as its domain, range, continuity, and behavior as x approaches positive or negative infinity. Each piece of the function contributes differently to its overall characteristics, and we will explore these contributions in detail. The exponential part, (1/2)^x + 3, governs the behavior for negative x values, while the quadratic part, -x^2 + 2, governs the behavior for non-negative x values. By examining these parts separately and then considering their connection at x = 0, we can gain a comprehensive understanding of the function. The piecewise nature of g(x) makes it an interesting subject for mathematical analysis, requiring a careful consideration of each piece and their interplay.

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In the case of function g(x), we need to consider both pieces of its definition. For x < 0, g(x) is defined as (1/2)^x + 3. Exponential functions like (1/2)^x are defined for all real numbers. Thus, this piece of the function is defined for all negative x-values. For x ≥ 0, g(x) is defined as -x^2 + 2. Quadratic functions like -x^2 + 2 are also defined for all real numbers. This piece of the function is defined for all non-negative x-values. Since the function is defined for all x < 0 and all x ≥ 0, it is defined for all real numbers. Therefore, the domain of g(x) is all real numbers, which can be written in interval notation as (-∞, ∞). This means that any real number can be input into the function, and the function will produce a corresponding output. The domain is an essential property of a function, as it tells us the set of values for which the function is meaningful. In the context of real-world applications, the domain might represent physical constraints or limitations on the input values. For g(x), the fact that its domain is all real numbers indicates that there are no such restrictions on the input values.

The range of a function refers to the set of all possible output values (y-values) that the function can produce. To determine the range of g(x), we need to analyze each piece of the function separately. For x < 0, g(x) = (1/2)^x + 3. The exponential function (1/2)^x approaches 0 as x approaches negative infinity and approaches infinity as x approaches 0. Therefore, (1/2)^x takes on all values greater than 0. Adding 3 to this result shifts the range up by 3, so this piece of the function takes on all values greater than 3. For x ≥ 0, g(x) = -x^2 + 2. This is a downward-opening parabola with a vertex at (0, 2). The maximum value of this piece is 2, which occurs at x = 0. As x moves away from 0, the values of -x^2 + 2 decrease, approaching negative infinity. Thus, this piece of the function takes on all values less than or equal to 2. Combining the ranges of both pieces, we have values greater than 3 and values less than or equal to 2. Therefore, the range of g(x) is (-∞, 2] ∪ (3, ∞). This means that the function can output any value in these intervals, but not values between 2 and 3. The range is another critical property of a function, as it indicates the possible output values. In practical scenarios, the range might represent the set of possible outcomes or results. For g(x), the disjointed range indicates a discontinuity or a gap in the output values.

Continuity is a fundamental property of functions in calculus. A function is continuous at a point if there is no break or jump in its graph at that point. For a piecewise function like g(x), we need to check continuity at the point where the definition changes, which is x = 0. To check continuity at x = 0, we need to verify three conditions: 1. g(0) exists. 2. The limit of g(x) as x approaches 0 exists. 3. The limit of g(x) as x approaches 0 is equal to g(0). Let's evaluate these conditions for g(x). g(0) is given by the second piece of the function definition: g(0) = -0^2 + 2 = 2. So, g(0) exists. Next, we need to find the limit of g(x) as x approaches 0. Since g(x) is defined piecewise, we need to consider the left-hand limit and the right-hand limit separately. The left-hand limit is the limit as x approaches 0 from the left (x < 0): lim (x→0-) g(x) = lim (x→0-) ((1/2)^x + 3) = (1/2)^0 + 3 = 1 + 3 = 4. The right-hand limit is the limit as x approaches 0 from the right (x ≥ 0): lim (x→0+) g(x) = lim (x→0+) (-x^2 + 2) = -0^2 + 2 = 2. Since the left-hand limit (4) and the right-hand limit (2) are not equal, the limit of g(x) as x approaches 0 does not exist. Therefore, g(x) is not continuous at x = 0. This discontinuity arises from the abrupt change in the function's definition at x = 0. The exponential piece approaches 4 as x approaches 0 from the left, while the quadratic piece is equal to 2 at x = 0. This jump in the function's value creates a discontinuity. The lack of continuity at x = 0 has implications for the function's behavior and its applicability in certain contexts. For instance, g(x) cannot be differentiated at x = 0 due to the discontinuity.

To determine which statements about function g(x) are true, we need to apply our understanding of the function's definition, domain, range, and continuity. Let's consider some example statements and how we would analyze them.

Example Statements:

1. "g(x) is continuous at x = 0." As we determined earlier, g(x) is not continuous at x = 0 because the left-hand limit and the right-hand limit at x = 0 are not equal. Therefore, this statement is false.
2. "The range of g(x) includes all real numbers." We found that the range of g(x) is (-∞, 2] ∪ (3, ∞). This range does not include the interval (2, 3]. Therefore, this statement is false.
3. "g(-1) = 5." To evaluate g(-1), we use the first piece of the function definition because -1 < 0: g(-1) = (1/2)^(-1) + 3 = 2 + 3 = 5. Therefore, this statement is true.
4. "g(1) = 1." To evaluate g(1), we use the second piece of the function definition because 1 ≥ 0: g(1) = -1^2 + 2 = -1 + 2 = 1. Therefore, this statement is true.
5. "g(x) has a maximum value of 2." The second piece of the function, -x^2 + 2, has a maximum value of 2 at x = 0. The first piece, (1/2)^x + 3, takes on values greater than 3 for x < 0. Therefore, the function does not have a maximum value of 2, this statement is false. It is crucial to carefully examine each statement in the context of the function's properties. By considering the function's piecewise definition and its behavior in different intervals, we can accurately assess the truthfulness of the statements.

In conclusion, analyzing a piecewise function like g(x) requires a thorough understanding of its definition, domain, range, and continuity. By examining each piece of the function separately and then considering their connection points, we can determine the function's key properties. We found that g(x) has a domain of all real numbers, a range of (-∞, 2] ∪ (3, ∞), and is discontinuous at x = 0. These properties are essential for evaluating statements about the function and understanding its behavior. The piecewise nature of g(x) makes it a valuable example for illustrating the complexities and nuances of function analysis. Understanding these concepts is crucial for success in calculus and related fields. The ability to analyze functions like g(x) is a fundamental skill for mathematicians, scientists, and engineers. It allows for the modeling and understanding of real-world phenomena that can be represented by piecewise functions. The careful consideration of domain, range, and continuity is essential for accurate analysis and interpretation. By mastering these concepts, one can gain a deeper appreciation for the power and versatility of mathematical functions.