Finding Domain And Range Of Transformed Functions G(x) = -f(1/4(x + 5))

In mathematics, functions are fundamental tools for modeling relationships between variables. Understanding the domain and range of a function is crucial for grasping its behavior and properties. The domain represents the set of all possible input values (x-values) for which the function is defined, while the range represents the set of all possible output values (y-values) that the function can produce. When we apply transformations to a function, such as reflections, stretches, and translations, the domain and range can change in predictable ways. This article delves into the process of determining the domain and range of a transformed function, specifically focusing on the function g(x) = -f(1/4(x + 5)), given the domain and range of the original function f(x).

Understanding the Original Function f(x)

Let's start by examining the original function, f(x). We are given that the domain of f(x) is [-1, 4], which means that f(x) is defined for all x-values between -1 and 4, inclusive. In other words, we can input any value from -1 to 4 into the function f, and it will produce a valid output. The range of f(x) is given as [6, 7], which indicates that the output values of f(x) will always fall between 6 and 7, inclusive. This information is essential as it serves as the foundation for understanding how transformations will affect the domain and range of the transformed function g(x).

To visualize this, imagine f(x) as a machine that takes inputs from -1 to 4 and produces outputs between 6 and 7. The transformations we apply will essentially alter how this machine processes inputs and generates outputs. Understanding the original domain and range provides a baseline for predicting these changes. We know the boundaries within which f(x) operates, and this knowledge will guide us as we analyze the transformed function.

Furthermore, it's important to recognize that the original domain and range define the limits of the function's behavior. Without this information, it would be impossible to accurately determine the domain and range of the transformed function. The transformations we'll discuss will stretch, shift, and reflect the function, but the original domain and range provide the anchor points for these changes. Therefore, a clear understanding of the initial domain and range is paramount for solving the problem at hand.

Analyzing the Transformation g(x) = -f(1/4(x + 5))

The function g(x) is a transformation of f(x), specifically defined as g(x) = -f(1/4(x + 5)). This transformation involves several key operations that will affect both the domain and range of the original function. To understand the changes, we need to break down the transformation step-by-step.

The transformation can be deconstructed into three main components: a horizontal shift, a horizontal stretch/compression, and a vertical reflection. Let's analyze each of these in detail:

1. Horizontal Shift: The term (x + 5) inside the function f indicates a horizontal shift. Specifically, it shifts the graph of f(x) to the left by 5 units. This means that to get the same output from g as we would from f, we need to input a value that is 5 units smaller. This shift will directly impact the domain of the transformed function.

2. Horizontal Stretch/Compression: The factor of 1/4 multiplying x inside the function f represents a horizontal stretch. A factor between 0 and 1 stretches the graph horizontally, while a factor greater than 1 compresses it. In this case, the 1/4 stretches the graph horizontally by a factor of 4. This means the domain will be expanded by a factor of 4 after the shift is applied.

3. Vertical Reflection: The negative sign in front of f indicates a vertical reflection across the x-axis. This means that the output values of f will be multiplied by -1, effectively flipping the graph upside down. This reflection will primarily affect the range of the transformed function.

By carefully analyzing each component of the transformation, we can predict how the original domain and range of f(x) will be altered to produce the domain and range of g(x). Understanding these individual effects is crucial for accurately determining the final domain and range of the transformed function.

Determining the Domain of g(x)

The domain of g(x) = -f(1/4(x + 5)) is affected by the horizontal transformations applied to the original function f(x). The domain of f(x) is given as [-1, 4]. To find the domain of g(x), we need to reverse the transformations applied to the x-values.

First, consider the horizontal stretch by a factor of 4 represented by the term 1/4 inside the function. To undo this stretch, we need to consider how the original domain boundaries are affected. The original domain [-1, 4] implies that: -1 ≤ x ≤ 4. Since the input to f in g(x) is (1/4)(x + 5), we need to find the values of x that make this expression fall within the domain of f. Therefore, we set up the following inequality:

-1 ≤ (1/4)(x + 5) ≤ 4

To solve this inequality, we first multiply all parts by 4:

-4 ≤ x + 5 ≤ 16

Next, we subtract 5 from all parts:

-4 - 5 ≤ x ≤ 16 - 5

-9 ≤ x ≤ 11

This inequality defines the domain of g(x). The horizontal shift and stretch have transformed the original domain [-1, 4] into [-9, 11]. Thus, the domain of g(x) in interval notation is [-9, 11]. This means that g(x) is defined for all x-values between -9 and 11, inclusive. Understanding how the horizontal transformations affect the domain is crucial for accurately defining the set of possible input values for the transformed function.

Determining the Range of g(x)

The range of g(x) = -f(1/4(x + 5)) is primarily affected by the vertical transformations applied to the original function f(x). The range of f(x) is given as [6, 7]. The key vertical transformation in this case is the reflection across the x-axis, represented by the negative sign in front of f.

Since g(x) = -f(1/4(x + 5)), the output values of g(x) will be the negative of the output values of f(1/4(x + 5)). This vertical reflection will flip the range of f(x) across the x-axis. To determine the new range, we simply multiply the original range boundaries by -1.

The original range of f(x) is [6, 7], which means that: 6 ≤ f(x) ≤ 7. When we apply the negative sign, the inequality becomes:

-7 ≤ -f(1/4(x + 5)) ≤ -6

This inequality indicates that the range of g(x) is between -7 and -6, inclusive. Therefore, the range of g(x) in interval notation is [-7, -6]. The vertical reflection has transformed the original range [6, 7] into [-7, -6]. It's important to note that the horizontal transformations do not affect the range of the function; only vertical transformations do. By understanding how the vertical reflection impacts the output values, we can accurately define the range of the transformed function.

Conclusion

In summary, given the function g(x) = -f(1/4(x + 5)), where the domain of f(x) is [-1, 4] and the range is [6, 7], we have determined the following:

• Domain of g(x): [-9, 11]
• Range of g(x): [-7, -6]

Understanding how transformations affect the domain and range of functions is a fundamental concept in mathematics. By carefully analyzing the transformations applied to a function, we can accurately predict how its domain and range will change. In this case, the horizontal shift and stretch affected the domain, while the vertical reflection affected the range. These techniques are essential for working with functions and their transformations in various mathematical contexts.