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 has significant applications in various fields, including calculus, linear algebra, and computer science. This article delves into the process of finding the inverse of the function y = 2x² + 4, exploring the steps involved, the challenges presented by quadratic functions, and the implications of the result. We'll also discuss the domain and range restrictions necessary to ensure a valid inverse function exists.

Understanding Inverse Functions

Before tackling the specific function, let's establish a clear understanding of inverse functions. An inverse function, denoted as f⁻¹(x), essentially "undoes" the operation of the original function f(x). If you apply a function to a value and then apply its inverse to the result, you should get back the original value. Mathematically, this is expressed as:

f⁻¹(f(x)) = x and f(f⁻¹(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 does not have an inverse function over its entire domain.

The Challenges with Quadratic Functions

The function y = 2x² + 4 is a quadratic function. Quadratic functions, in their general form, are not one-to-one. This is because a parabola (the graph of a quadratic function) is symmetrical about its vertex. A horizontal line drawn above the vertex will intersect the parabola at two points, violating the one-to-one condition.

To overcome this limitation and find an inverse, we need to restrict the domain of the original function. We'll choose a portion of the parabola where it passes the horizontal line test – typically, either the left or right half of the parabola. This restriction creates a one-to-one relationship between the input and output values.

Finding the Inverse: Step-by-Step Process

Let's outline the steps to find the inverse of y = 2x² + 4, remembering the necessary domain restriction:

Step 1: Swap x and y

The first step in finding the inverse is to swap the variables x and y in the original function equation. This gives us:

x = 2y² + 4

Step 2: Solve for y

Now, we need to solve this equation for y. This involves a series of algebraic manipulations:

• Subtract 4 from both sides: x - 4 = 2y²
• Divide both sides by 2: (x - 4) / 2 = y²
• Take the square root of both sides: y = ±√[(x - 4) / 2]

Notice the ± sign. This highlights the fact that for a single x value, there are two corresponding y values, further emphasizing the non-one-to-one nature of the unrestricted quadratic function.

Step 3: Apply Domain Restriction

To obtain a valid inverse function, we must restrict the domain of the original function. Let's choose the non-negative portion of the parabola, restricting the domain of the original function to x ≥ 0. This corresponds to the right half of the parabola. This restriction forces us to select only the positive square root:

y = √[(x - 4) / 2]

This is the inverse function for the restricted domain of the original function.

Step 4: Verify the Inverse (Optional but Recommended)

It's always a good practice to verify that the function we've found is indeed the inverse. We can do this by checking the properties mentioned earlier:

f⁻¹(f(x)) = x and f(f⁻¹(x)) = x

Let's check:

• f⁻¹(f(x)): Substitute f(x) = 2x² + 4 into the inverse function: √[((2x² + 4) - 4) / 2] = √(2x²/2) = √x² = x (Since x ≥ 0 in our restricted domain)

• f(f⁻¹(x)): Substitute f⁻¹(x) = √[(x - 4) / 2] into the original function: 2 * (√[(x - 4) / 2])² + 4 = 2 * [(x - 4) / 2] + 4 = x - 4 + 4 = x

Both conditions are satisfied, confirming that y = √[(x - 4) / 2] is the inverse function of y = 2x² + 4 for x ≥ 0.

Domain and Range of the Original Function and its Inverse

It's crucial to understand the domain and range of both the original function and its inverse.

Original Function (y = 2x² + 4):

• Domain: (-∞, ∞) (All real numbers)
• Range: [4, ∞) (All real numbers greater than or equal to 4)

Inverse Function (y = √[(x - 4) / 2]):

• Domain: [4, ∞) (This is the range of the original function and the restricted domain)
• Range: [0, ∞) (This is the positive portion of the domain of the original function)

Graphical Representation

Graphing both the original function and its inverse helps visualize the relationship and the effect of the domain restriction. The graph of the inverse function will be the reflection of the restricted portion of the original function's graph about the line y = x.

Applications of Inverse Functions

Inverse functions have wide-ranging applications across various fields:

• Cryptography: Encryption and decryption algorithms often rely on inverse functions.
• Computer Science: Data transformation and encoding techniques utilize inverse functions.
• Calculus: Finding derivatives and integrals often involves working with inverse functions.
• Economics: Modeling supply and demand relationships sometimes uses inverse functions.

Conclusion

Finding the inverse of a quadratic function like y = 2x² + 4 requires careful consideration of the function's properties and the application of domain restrictions to ensure the inverse function is well-defined. The process involves swapping variables, solving for y, and carefully choosing a restricted domain to make the function one-to-one. By understanding these steps and their implications, one can confidently tackle similar problems and appreciate the significance of inverse functions in various mathematical and applied contexts. Remember to always verify your solution by checking the properties of inverse functions: f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. This verification step is crucial in confirming the accuracy of your calculations and understanding the relationship between the original function and its inverse. Finally, paying close attention to the domain and range of both the original function and its inverse is essential for a complete and accurate understanding of the problem.