Domain And Range Of W(x) = -(3x)^(1/2) - 4 A Comprehensive Guide

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In the realm of mathematical functions, transformations play a crucial role in shaping and altering the behavior of parent functions. Understanding these transformations is essential for predicting the domain and range of the resulting functions. In this article, we will delve into the transformation of the square root function, f(x) = x^(1/2), to a new function, w(x) = -(3x)^(1/2) - 4. Our primary focus will be on determining the domain and range of this transformed function, w(x). We will break down the transformations step by step, analyze their impact on the original domain and range, and arrive at the final domain and range for w(x). This exploration will not only enhance our understanding of function transformations but also reinforce the fundamental concepts of domain and range.

Before we embark on the transformation journey, it's imperative to have a firm grasp of the parent function, f(x) = x^(1/2), also known as the square root function. This function forms the bedrock for our transformation, and its inherent properties will influence the characteristics of the transformed function. Let's begin by dissecting the domain and range of f(x) = x^(1/2).

Domain of f(x) = x^(1/2):

The domain of a function encompasses all possible input values (x-values) for which the function produces a real output. For the square root function, the radicand (the expression under the square root) must be non-negative to yield a real number. Mathematically, this translates to x ≥ 0. Therefore, the domain of f(x) = x^(1/2) is [0, ∞), indicating that the function accepts all non-negative real numbers as input.

Range of f(x) = x^(1/2):

The range of a function represents the set of all possible output values (y-values or f(x)-values) that the function can produce. Since the square root of a non-negative number is always non-negative, the output of f(x) = x^(1/2) is also non-negative. Thus, the range of f(x) = x^(1/2) is [0, ∞), signifying that the function generates all non-negative real numbers as output.

In summary, the parent function f(x) = x^(1/2) has a domain of [0, ∞) and a range of [0, ∞). These characteristics will serve as our reference point as we analyze the transformations applied to this function.

Now, let's turn our attention to the transformed function, w(x) = -(3x)^(1/2) - 4. This function is a modified version of the parent function, f(x) = x^(1/2), and understanding the transformations applied is key to determining its domain and range. We can break down the transformation into three distinct steps:

  1. Horizontal Compression: The term 3x inside the square root indicates a horizontal compression. The graph of y = (3x)^(1/2) is compressed horizontally by a factor of 1/3 compared to the graph of y = x^(1/2). This compression affects the domain of the function.

  2. Reflection across the x-axis: The negative sign in front of the square root, -(3x)^(1/2), signifies a reflection across the x-axis. This reflection flips the graph vertically, impacting the range of the function. The positive outputs of the square root function become negative.

  3. Vertical Shift: The subtraction of 4, -(3x)^(1/2) - 4, represents a vertical shift downwards by 4 units. This shift moves the entire graph down, directly affecting the range of the function. All output values are decreased by 4.

By carefully analyzing each transformation, we can predict how the domain and range of the parent function will be altered. The horizontal compression will affect the domain, the reflection across the x-axis will invert the range, and the vertical shift will lower the range.

To determine the domain of w(x) = -(3x)^(1/2) - 4, we need to consider the input values (x-values) that produce real outputs. As with the parent function, the expression under the square root must be non-negative. This leads us to the inequality:

3x ≥ 0

Dividing both sides by 3, we get:

x ≥ 0

This inequality reveals that the domain of w(x) consists of all non-negative real numbers. In interval notation, this is expressed as:

Domain of w(x): [0, ∞)

Notice that the horizontal compression by a factor of 1/3 does not change the lower bound of the domain, which remains 0. This is because compressing the graph horizontally only squeezes it along the x-axis but doesn't shift the starting point of the function.

Determining the range of w(x) = -(3x)^(1/2) - 4 requires us to trace the impact of the transformations on the original range of the parent function, f(x) = x^(1/2), which is [0, ∞). Let's analyze each transformation step by step:

  1. (3x)^(1/2): This horizontal compression does not affect the range. The range remains [0, ∞).

  2. -(3x)^(1/2): The reflection across the x-axis inverts the range. The non-negative outputs become non-positive, transforming the range from [0, ∞) to (-∞, 0].

  3. -(3x)^(1/2) - 4: The vertical shift downwards by 4 units shifts the entire range down by 4. This transforms the range from (-∞, 0] to (-∞, -4].

Therefore, the range of w(x) = -(3x)^(1/2) - 4 is (-∞, -4]. This indicates that the function produces all real numbers less than or equal to -4.

In summary, the range of the transformed function w(x) is significantly different from the parent function due to the reflection and vertical shift. The function now outputs only negative values, bounded above by -4.

In this comprehensive exploration, we have successfully determined the domain and range of the transformed function w(x) = -(3x)^(1/2) - 4. By carefully dissecting the transformations applied to the parent function, f(x) = x^(1/2), we identified the horizontal compression, reflection across the x-axis, and vertical shift. These transformations collectively shaped the domain and range of w(x).

We found that the domain of w(x) is [0, ∞), signifying that the function accepts all non-negative real numbers as input. The range of w(x) is (-∞, -4], indicating that the function produces all real numbers less than or equal to -4 as output.

This analysis underscores the importance of understanding function transformations in predicting the behavior of functions. By grasping the effects of these transformations on the domain and range, we can gain valuable insights into the characteristics of various functions and their graphical representations. The skills and knowledge acquired in this exploration will undoubtedly prove beneficial in tackling more complex mathematical problems involving function transformations.

  • Domain of w(x): x ≥ 0 or [0, ∞)
  • Range of w(x): w(x) ≤ -4 or (-∞, -4]