Find the inverse of the function $y=2 x^2+2$.
To find the inverse of the function \(y = 2x^2 + 2\), we begin by understanding that an inverse relation is found by swapping the roles of the input variable \(x\) and the output variable \(y\). We start with the original function:
\[y = 2x^2 + 2\]
Swap the variables \(x\) and \(y\) in the original equation to represent the inverse relation:
\[x = 2y^2 + 2\]
Now, we solve for \(y\) to express it in terms of \(x\). First, subtract 2 from both sides of the equation to isolate the term containing \(y\):
\[x - 2 = 2y^2\]
Next, divide both sides by 2 to isolate \(y^2\):
\[\frac{x - 2}{2} = y^2\]
This can be rewritten as:
\[y^2 = \frac{x}{2} - \frac{2}{2} = \frac{1}{2}x - 1\]
Finally, take the square root of both sides to solve for \(y\). Since the square root can result in both a positive and a negative value, we include the plus-or-minus sign:
\[y = \pm \sqrt{\frac{1}{2}x - 1}\]
Comparing this result with the provided options, we see that it matches option b.
The concept of inverse functions is widely used in fields like computer graphics, where you might need to reverse a transformation to recover the original coordinates of an object. For instance, if a function maps a 2D point to a transformed position, its inverse allows you to "undo" the transformation. Understanding how to solve for the inverse, even for non-linear functions like quadratics, is essential for these types of coordinate systems, data recovery, and error correction algorithms.