Inverse Of Quadratic Function F(x) = -2(x+5)^2 + 6 And Domain Restrictions

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In the realm of mathematics, exploring the inverses of functions unveils fascinating relationships and properties. This article delves into the intricacies of finding the inverse of the quadratic function f(x) = -2(x+5)² + 6, and further investigates the conditions under which the inverse itself qualifies as a function. We will address the key aspects of determining the inverse equation without resorting to the quadratic formula and delve into the necessary domain restrictions to ensure the inverse adheres to the functional criteria. This exploration offers valuable insights into the behavior of quadratic functions and their inverses, enriching our understanding of mathematical transformations and their implications.

a) Determining the Equation of the Inverse of f(x) = -2(x+5)² + 6

To determine the inverse equation of the given quadratic function f(x) = -2(x+5)² + 6, we embark on a step-by-step process that unveils the underlying relationship between the original function and its inverse. The core principle behind finding an inverse lies in swapping the roles of the input (x) and output (y) variables, effectively reversing the function's operation. This reversal allows us to express the original input in terms of the original output, thereby defining the inverse function.

Our starting point is the function's equation, where f(x) represents the output, often denoted as y. So, we have y = -2(x+5)² + 6. The first crucial step is to interchange x and y to reflect the inverse relationship. This yields the equation x = -2(y+5)² + 6. Now, our objective is to isolate y on one side of the equation, which will express y as a function of x, effectively defining the inverse function, denoted as f⁻¹(x).

The journey to isolate y involves a series of algebraic manipulations. First, we subtract 6 from both sides of the equation, resulting in x - 6 = -2(y+5)². This step isolates the term containing the squared expression. Next, we divide both sides by -2 to further simplify the equation, obtaining (x - 6) / -2 = (y+5)². To eliminate the square, we take the square root of both sides, remembering to consider both the positive and negative roots. This leads to ±√((x - 6) / -2) = y + 5. It's essential to recognize that the presence of both positive and negative roots signifies that the inverse, as it stands, may not be a function, a point we'll delve into later.

Finally, to completely isolate y, we subtract 5 from both sides, giving us y = -5 ± √((x - 6) / -2). This equation represents the inverse of the original function. However, the ± sign indicates that for a single x value, there could be two corresponding y values, which violates the definition of a function. This highlights the significance of considering domain restrictions when dealing with inverses of quadratic functions. The expression inside the square root, (x - 6) / -2, must be non-negative for the square root to be a real number, which imposes a restriction on the domain of the inverse function. This restriction will be crucial in determining whether the inverse is a function and, if not, how to restrict the domain of the original function to make the inverse a function.

In summary, the equation we derived, y = -5 ± √((x - 6) / -2), represents the inverse of the function f(x) = -2(x+5)² + 6. However, the presence of the ± sign and the square root necessitates a closer examination of whether this inverse is indeed a function and, if not, what domain restrictions are required to ensure it behaves as a function.

b) Is the Inverse of f(x) a Function? Restricting the Domain of f(x)

The critical question at this juncture is: Is the inverse of f(x) = -2(x+5)² + 6 a function? To answer this, we must recall the fundamental definition of a function: for every input value (x), there must be only one corresponding output value (y). In the context of the inverse we derived, y = -5 ± √((x - 6) / -2), the presence of the ± sign immediately raises concerns. For a single x value, we potentially have two y values, one from the positive square root and another from the negative square root. This directly contradicts the definition of a function.

Therefore, the inverse as it stands is not a function. The reason lies in the nature of the original quadratic function. Quadratic functions, due to their parabolic shape, fail the horizontal line test. This test states that if any horizontal line intersects the graph of a function more than once, its inverse is not a function. The parabola opens downwards in this case because of the negative coefficient (-2) in front of the squared term, and its vertex is at (-5, 6). This means that for any y-value less than 6, there are two corresponding x-values, leading to the non-functional inverse.

To rectify this situation, we must restrict the domain of the original function f(x). The key is to limit the domain such that the function becomes one-to-one, meaning each x-value maps to a unique y-value. This can be achieved by considering only one half of the parabola, either the left side or the right side of the vertex. The vertex of the parabola plays a crucial role in determining the restriction. The x-coordinate of the vertex, which is -5, serves as the dividing line for the domain restriction.

We have two options for restricting the domain: either x ≥ -5 or x ≤ -5. Let's analyze each option. If we choose x ≥ -5, we are considering the right side of the parabola. In this region, the function is strictly decreasing, and each x-value corresponds to a unique y-value. Alternatively, if we choose x ≤ -5, we are considering the left side of the parabola. In this region, the function is strictly increasing, and again, each x-value corresponds to a unique y-value. Either restriction will make the original function one-to-one, ensuring that its inverse is also a function.

However, the choice of restriction will affect the specific form of the inverse function. If we restrict the domain to x ≥ -5, we consider only the negative square root in the inverse equation, resulting in f⁻¹(x) = -5 - √((6 - x) / 2). This ensures that the inverse function maps each x-value to a unique y-value within the restricted domain. Conversely, if we restrict the domain to x ≤ -5, we consider only the positive square root, leading to f⁻¹(x) = -5 + √((6 - x) / 2).

In conclusion, the inverse of f(x) = -2(x+5)² + 6 is not a function in its unrestricted form. To make the inverse a function, we must restrict the domain of the original function to either x ≥ -5 or x ≤ -5. The choice of restriction determines the specific form of the inverse function, ensuring that it adheres to the fundamental definition of a function. This domain restriction is a crucial aspect of working with inverses of quadratic functions, highlighting the importance of understanding the underlying properties of functions and their transformations.

By restricting the domain, we essentially carve out a portion of the original quadratic function that is one-to-one, allowing us to define a true inverse function. This inverse function will then map the range of the restricted original function back to its restricted domain, creating a clear and unambiguous relationship between input and output values. The process of domain restriction underscores the significance of considering the functional criteria when dealing with inverses and provides a valuable tool for manipulating functions to achieve desired properties.

Inverse Function, Quadratic Function, Domain Restriction, One-to-One Function, Horizontal Line Test, Vertex, Function Inverses, Mathematical Transformations, Algebraic Manipulations, Square Root, Real Number, Functional Criteria