Find the inverse of the function.
$y=2 x^2-4$
To find the inverse of the function \(y = 2x^2 - 4\), we first recognize that the function is a parabola. To find an inverse-like expression, we swap the roles of \(x\) and \(y\). The given relationship is \(y = 2x^2 - 4\). Swapping \(x\) and \(y\) gives the equation \(x = 2y^2 - 4\). Our objective is to solve this equation for \(y\).
We start with the equation \(x = 2y^2 - 4\). To isolate the term with \(y\), we first add \(4\) to both sides:
\[x + 4 = 2y^2\]
Next, divide both sides by \(2\) to isolate \(y^2\):
\[y^2 = \frac{x + 4}{2}\]
Finally, to solve for \(y\), take the square root of both sides. Since \(y^2 = c\) implies \(y = \pm \sqrt{c}\), we obtain:
\[y = \pm \sqrt{\frac{x + 4}{2}}\]
This expression shows the two possible branches of the relation that corresponds to the inverse of the original function over its restricted domains.
Understanding inverse functions is crucial in fields like physics and engineering, where you often need to reverse a process to find an original input from an output. For example, if a machine's output \(y\) is related to the input \(x\) by a specific formula, finding the inverse allows an operator to determine what input setting \(x\) was used to produce a measured result \(y\). Knowing how to manipulate these equations algebraically is the foundation for solving such real-world inverse problems.