find the inverse of the function y = 2x2 + 2.

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31-03-2025

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Finding the inverse of a function essentially means reversing its operation. If the original function takes an input (x) and produces an output (y), the inverse function takes that output (y) and produces the original input (x). However, not all functions have an inverse. For a function to have an inverse, it must be one-to-one (also called injective), meaning each input maps to a unique output. The function y = 2x² + 2 is not one-to-one because both x = 2 and x = -2 would produce the same output (y = 10). This means we can only find a partial inverse, and we'll need to restrict the domain.

Understanding One-to-One Functions and Inverse Functions

A function is one-to-one if it passes 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. The graph of y = 2x² + 2 is a parabola, and any horizontal line above y = 2 will intersect it twice. Therefore, to find an inverse, we must restrict the domain. Let's restrict the domain to x ≥ 0 (non-negative values). This part of the parabola passes the horizontal line test.

Step-by-Step Process to Find the Partial Inverse

Here's how to find the inverse function, assuming x ≥ 0:

1. Swap x and y: This is the first step in finding the inverse of any function. We swap the variables x and y in the original equation:

   x = 2y² + 2

2. Solve for y: Now we need to solve this new equation for y. This is where the algebra comes in.

   x - 2 = 2y² (x - 2) / 2 = y² y = ±√[(x - 2) / 2]

3. Consider the Domain Restriction: Remember our domain restriction x ≥ 0? This means we are only considering the right half of the parabola. Since this half is increasing, the inverse will also be increasing. Therefore, we take only the positive square root:

   y = √[(x - 2) / 2]

4. Specify the Range: The original function had a range of y ≥ 2. Because we’ve found the inverse, the domain of the inverse function is x ≥ 2 and the range is y ≥ 0.

The Inverse Function

Therefore, the inverse function for y = 2x² + 2, with the domain restricted to x ≥ 0, is:

f⁻¹(x) = √[(x - 2) / 2], x ≥ 2

Visualizing the Inverse

Graphing both the original function and its inverse can be helpful. You'll notice that the graph of the inverse is a reflection of the original function (restricted to x ≥ 0) across the line y = x.

Why Restricting the Domain is Crucial

Without restricting the domain, we wouldn't have a true inverse function. The resulting equation y = ±√[(x - 2) / 2] gives two y-values for each x-value, which violates the definition of a function.

Practical Applications

Finding inverse functions has many applications in various fields, including mathematics, physics, and computer science. For example, it is crucial in solving equations and transforming data.

Remember, the process of finding the inverse involves swapping x and y, solving for y, and carefully considering any domain restrictions. If a horizontal line intersects the graph more than once, the function isn't one-to-one and doesn't have a true inverse without restricting the domain.