Find The Inverse Of The Function Y 2x2 4

Finding the Inverse of the Function y = 2x² + 4

Finding the inverse of a function is a fundamental concept in algebra and calculus. It involves switching the roles of the independent and dependent variables, essentially reversing the operation of the original function. While some functions have straightforward inverses, others, like the quadratic function y = 2x² + 4, present a slightly more complex challenge. This article will delve deep into the process of finding the inverse of this specific function, explaining the concepts involved and highlighting important considerations.

Understanding Inverse Functions

Before we tackle the inverse of y = 2x² + 4, let's establish a solid understanding of inverse functions in general. An inverse function, denoted as f⁻¹(x), "undoes" the action of the original function f(x). This means that if you apply a function and then its inverse (or vice versa), you'll end up back where you started. Mathematically, this is expressed as:

• f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

This property only holds true if the original function is one-to-one (also known as injective). A one-to-one function means that each input value (x) maps to a unique output value (y), and vice-versa. Graphically, this is represented by the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one and doesn't have a true inverse over its entire domain.

The Challenge with Quadratic Functions

Our function, y = 2x² + 4, is a quadratic function. Quadratic functions are inherently not one-to-one over their entire domain (all real numbers). This is because for any positive y-value, there are two corresponding x-values (one positive and one negative) that satisfy the equation. This is evident in the parabolic shape of the graph, where a horizontal line would intersect the parabola at two points.

To find an inverse, we must restrict the domain of the original function to make it one-to-one. This means we'll only consider a portion of the parabola where the horizontal line test passes. The most common approach is to restrict the domain to either x ≥ 0 (the right half of the parabola) or x ≤ 0 (the left half). We will choose x ≥ 0 for this example.

Finding the Inverse: Step-by-Step Process

Now, let's proceed with finding the inverse of y = 2x² + 4, restricting the domain to x ≥ 0:

Step 1: Swap x and y:

This is the first crucial step in finding the inverse. We exchange the places of x and y in the original equation:

x = 2y² + 4

Step 2: Solve for y:

Now, our goal is to isolate y in the equation. This involves a series of algebraic manipulations:

1. Subtract 4 from both sides: x - 4 = 2y²

2. Divide both sides by 2: (x - 4) / 2 = y²

3. Take the square root of both sides: y = ±√[(x - 4) / 2]

Step 3: Consider the Restricted Domain:

Remember, we restricted the domain of the original function to x ≥ 0. This restriction carries over to the inverse function. Since we chose the positive branch of the parabola (x ≥ 0), we only consider the positive square root to maintain consistency:

y = √[(x - 4) / 2]

Step 4: Specify the Domain and Range of the Inverse:

The domain of the inverse function is the range of the original function (with the restriction), and the range of the inverse function is the domain of the restricted original function.

• Domain of the inverse function: The range of y = 2x² + 4 where x ≥ 0 is y ≥ 4. Therefore, the domain of the inverse is x ≥ 4.

• Range of the inverse function: The domain of the restricted original function is x ≥ 0. Therefore, the range of the inverse is y ≥ 0.

Therefore, the inverse function is:

f⁻¹(x) = √[(x - 4) / 2], for x ≥ 4

Verification: Checking the Inverse

To ensure our calculated inverse is correct, we should verify it using the properties of inverse functions: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Verification 1: f(f⁻¹(x)) = x

1. Substitute f⁻¹(x) into f(x): f(f⁻¹(x)) = 2[√((x - 4) / 2)]² + 4

2. Simplify: f(f⁻¹(x)) = 2[(x - 4) / 2] + 4 = x - 4 + 4 = x

Verification 2: f⁻¹(f(x)) = x

1. Substitute f(x) into f⁻¹(x): f⁻¹(f(x)) = √[(2x² + 4 - 4) / 2]

2. Simplify: f⁻¹(f(x)) = √[(2x²) / 2] = √(x²) = |x|

Since we restricted the domain of f(x) to x ≥ 0, |x| simplifies to x, confirming the inverse relationship.

Graphical Representation

Graphing both the original function (y = 2x² + 4, x ≥ 0) and its inverse (y = √[(x - 4) / 2], x ≥ 4) will visually demonstrate their inverse relationship. You'll observe that the graph of the inverse is a reflection of the original function across the line y = x. This reflection is a hallmark characteristic of inverse functions.

Implications and Applications

Understanding how to find the inverse of functions, especially those requiring domain restrictions, is crucial in various mathematical and scientific fields. Applications include:

• Cryptography: Inverse functions play a vital role in encryption and decryption algorithms.

• Calculus: Finding the inverse is essential for calculating derivatives and integrals of certain functions.

• Economics: Inverse functions are used in supply and demand models.

Conclusion

Finding the inverse of y = 2x² + 4 requires careful consideration of the function's properties and the application of appropriate domain restrictions. By systematically following the steps outlined in this article – swapping x and y, solving for y, considering the restricted domain, and verifying the result – you can confidently determine the inverse of this and similar quadratic functions. Remember, the key lies in understanding the concept of one-to-one functions and the significance of domain restrictions in ensuring a valid inverse exists. The inverse function we derived, √[(x - 4) / 2] for x ≥ 4, accurately reflects the inverse relationship with the restricted original function, providing a solid foundation for further applications.