The Graph Of F(x)=√x Reflected Across The Y-Axis Understanding Domain Comparisons

Introduction to Function Transformations

In the fascinating world of mathematics, understanding how functions transform is crucial for grasping the behavior and characteristics of various equations. Transformations, such as reflections, stretches, and shifts, alter the graph of a function, leading to new functions with different properties. Among these properties, the domain of a function—the set of all possible input values (x-values) for which the function is defined—is a fundamental concept. In this article, we will delve into the specific transformation of reflecting the graph of the function f(x) = √x across the y-axis and explore how this reflection affects its domain.

The domain of a function is a critical aspect that dictates the values we can input into the function without causing it to be undefined. For square root functions like f(x) = √x, the domain is restricted to non-negative numbers because the square root of a negative number is not defined within the realm of real numbers. This restriction significantly impacts how we interpret and manipulate these functions, especially when transformations come into play. Reflections, in particular, can dramatically alter the domain by mirroring the graph across an axis, potentially opening up new input values while restricting others.

Our focus in this discussion is the reflection of f(x) = √x across the y-axis. This transformation mirrors the graph horizontally, which intuitively suggests that the domain might change. To fully understand this, we'll need to analyze the original domain of f(x) = √x, visualize the reflection, and then determine the new domain of the reflected function, which we'll call g(x). This exploration will not only enhance our understanding of function transformations but also reinforce the importance of domains in defining the behavior of functions. We will meticulously examine each step, providing clear explanations and examples to ensure a comprehensive understanding of the concepts involved. By the end of this article, you will be well-equipped to analyze similar transformations and their effects on the domains of various functions.

The Original Function: f(x) = √x and Its Domain

Before we can explore the effects of reflection, it's essential to thoroughly understand the original function, f(x) = √x. The square root function, denoted as √x, is a fundamental mathematical operation that returns a value which, when multiplied by itself, yields x. For instance, the square root of 9 is 3 because 3 * 3 = 9. However, there's a crucial restriction: we cannot take the square root of a negative number within the set of real numbers because there is no real number that, when multiplied by itself, results in a negative value. This restriction is the cornerstone of the domain of f(x) = √x.

The domain of a function is defined as the set of all possible input values (x-values) for which the function produces a real number output. In the case of f(x) = √x, the input x must be greater than or equal to 0. If x were negative, the square root would result in an imaginary number, which falls outside the scope of real number outputs. Therefore, the domain of f(x) = √x is all non-negative real numbers, which can be expressed mathematically as x ≥ 0. This means that we can input any number that is zero or positive into the function, and it will yield a real number output.

Visually, the graph of f(x) = √x starts at the origin (0,0) and extends to the right, increasing gradually as x increases. This graphical representation directly reflects the domain restriction; there is no part of the graph to the left of the y-axis, which corresponds to negative x-values. The function is only defined for x-values on and to the right of the y-axis. Understanding this graphical representation is crucial because it allows us to visualize how transformations, such as reflections, will alter the domain. The shape of the graph and its position relative to the axes give us a clear picture of the function's behavior and its limitations. Therefore, before we consider reflecting the graph, it's imperative to firmly grasp the original function's domain and its visual representation, setting the stage for a comprehensive analysis of the transformed function.

Reflecting Across the Y-Axis: Creating g(x)

Now that we have a solid understanding of f(x) = √x and its domain, we can explore the transformation of reflecting this function across the y-axis. A reflection across the y-axis is a geometric transformation that mirrors a graph over the vertical axis. This means that every point (x, y) on the original graph is transformed to a new point (-x, y) on the reflected graph. In simpler terms, the x-coordinate changes its sign while the y-coordinate remains the same. This horizontal flip has significant implications for the function's domain, as it essentially reverses the set of allowable x-values.

To create the function g(x), which is the reflection of f(x) = √x across the y-axis, we need to replace x in the original function with -x. This is because the transformation (x, y) → (-x, y) mathematically represents the reflection. Thus, g(x) = √(-x). This new function looks similar to the original but has a crucial difference: the input to the square root is now -x instead of x. This seemingly small change has a profound impact on the domain of the function.

Visualizing this transformation can be extremely helpful. Imagine the graph of f(x) = √x being mirrored across the y-axis. The part of the graph that was on the right side of the y-axis (where x is positive) now appears on the left side (where x is negative), and vice versa. The point (1, 1) on f(x), for example, becomes (-1, 1) on g(x). The key insight here is that the reflected graph now extends into the negative x-values, which were previously excluded from the domain of f(x). This reflection fundamentally alters the set of permissible inputs, making it necessary to re-evaluate the domain. The reflected function g(x) = √(-x) is the mirror image of f(x) = √x, and understanding this visual transformation is crucial for determining how the domain changes.

Comparing the Domains of f(x) and g(x)

With the reflection across the y-axis transforming f(x) = √x into g(x) = √(-x), the crucial question now is: how does this transformation affect the domain? We know that the domain of f(x) = √x is x ≥ 0, meaning only non-negative numbers can be input into the function. However, the introduction of -x inside the square root in g(x) changes the game entirely. To ensure that g(x) produces a real number output, the expression inside the square root, -x, must be greater than or equal to zero. This is because, just like with f(x), we cannot take the square root of a negative number in the realm of real numbers.

Mathematically, we express this condition as -x ≥ 0. To solve this inequality for x, we can multiply both sides by -1. Remember, when multiplying or dividing an inequality by a negative number, we must reverse the inequality sign. Thus, we get x ≤ 0. This is the domain of g(x) = √(-x). It states that the permissible input values for g(x) are all real numbers that are less than or equal to zero. In other words, g(x) is defined for all non-positive numbers.

Comparing the domains of f(x) and g(x), we see a clear distinction. The domain of f(x) = √x is x ≥ 0, which includes all non-negative real numbers. The domain of g(x) = √(-x) is x ≤ 0, which includes all non-positive real numbers. The reflection across the y-axis has effectively flipped the domain from the non-negative side of the number line to the non-positive side. Visually, this means the graph of g(x) exists only on the left side of the y-axis and on the y-axis itself, whereas the graph of f(x) exists only on the right side and on the y-axis. This comparison highlights how reflections can significantly alter the domain of a function, and understanding these changes is crucial for analyzing the behavior and properties of transformed functions.

Conclusion: Key Differences in Domains

In conclusion, reflecting the graph of f(x) = √x across the y-axis to create g(x) = √(-x) significantly impacts the function's domain. The original function, f(x) = √x, has a domain of x ≥ 0, meaning it is defined for all non-negative real numbers. This is because the square root of a negative number is not a real number, and thus, only non-negative inputs are permissible.

The transformation of reflection across the y-axis results in the function g(x) = √(-x). This seemingly simple change has profound implications for the domain. For g(x) to be defined within the realm of real numbers, the expression inside the square root, -x, must also be non-negative. This leads to the condition -x ≥ 0, which, when solved for x, gives us x ≤ 0. Therefore, the domain of g(x) = √(-x) is x ≤ 0, meaning it is defined for all non-positive real numbers.

Comparing the domains of f(x) and g(x) reveals a clear and crucial difference. The domain of f(x) includes all non-negative numbers, while the domain of g(x) includes all non-positive numbers. This stark contrast underscores the effect of the reflection across the y-axis, which mirrors the function's graph and, consequently, its domain. The reflection effectively flips the set of allowable input values from the right side of the y-axis to the left side. This understanding is vital for anyone studying function transformations, as it highlights how geometric operations can significantly alter the fundamental properties of a function. The domain is a key characteristic of any function, and transformations such as reflections can dramatically change it, necessitating a careful analysis of the transformed function.

Therefore, the correct answer is:

C. The domain of f is x≥ 0, and the domain of g is x≤ 0.