Choose the inverse of $y=x^2-10 x$.
We are tasked with finding the inverse of the function \(y = x^2 - 10x\). To find the inverse of a function, we typically interchange the roles of \(x\) and \(y\) and then solve the resulting equation for \(y\). We begin by rewriting the given expression by completing the square to make it easier to manipulate.
The original function is \(y = x^2 - 10x\). To complete the square for \(x^2 - 10x\), we take half of the coefficient of \(x\), which is \(-10/2 = -5\), and square it to get \(25\). Thus, we can rewrite the expression as:
\[y = (x^2 - 10x + 25) - 25 = (x - 5)^2 - 25.\]
To find the inverse, we swap \(x\) and \(y\) in our equation:
\[x = (y - 5)^2 - 25.\]
Now, we solve for \(y\). First, add 25 to both sides:
\[x + 25 = (y - 5)^2.\]
Next, take the square root of both sides, which introduces a plus-or-minus sign because there are two potential values for \(y-5\):
\[\pm \sqrt{x + 25} = y - 5.\]
Finally, add 5 to both sides to isolate \(y\):
\[y = \pm \sqrt{x + 25} + 5.\]
Comparing our result \(y = \pm \sqrt{x + 25} + 5\) with the given options:
The correct option is (c).
Finding the inverse of a function is a fundamental concept used in various fields, such as economics, engineering, and physics, to reverse processes. For example, if a function \(f(t)\) models the distance traveled by a vehicle over time \(t\), its inverse \(f^{-1}(d)\) can be used to determine the time required to cover a specific distance \(d\). By mastering this process with quadratic expressions, you gain the ability to analyze dynamic systems and understand how to trace variables backward in time or space.