Which equation can be simplified to find the inverse of $y=5 x^2+10$?
Final Answer: \(\boxed{d}\)
The function given is \(y = 5x^2 + 10\). To find the inverse of a function \(y = f(x)\), the standard procedure is to swap the roles of \(x\) and \(y\). This means that every instance of \(y\) in the original equation is replaced by \(x\), and every instance of \(x\) is replaced by \(y\). The resulting equation describes the relationship between the new \(x\) and \(y\) values, which allows us to solve for \(y\) in terms of \(x\) to find the inverse function.
Applying this procedure to the original function \(y = 5x^2 + 10\), we replace \(y\) with \(x\) and \(x\) with \(y\). This transformation yields:
\[x = 5y^2 + 10\]
This equation now represents the relation that we need to manipulate in order to express \(y\) in terms of \(x\), which is the standard step to finding the inverse function.
We now compare the derived equation \(x = 5y^2 + 10\) to the provided options:
The equation \(x = 5y^2 + 10\) matches option d. This is the correct equation that, when solved for \(y\), gives the inverse relation.
Finding the inverse function is akin to reversing a process. Imagine a machine that takes an input, squares it, multiplies it by 5, and adds 10 to produce an output. Finding the inverse function is like creating a new machine that takes that output and reverses each step in the opposite order—subtracting 10, dividing by 5, and taking the square root—to recover the original input. This is fundamentally how data decryption and control systems work, where you need to undo a transformation to return to the initial state.