Final answer:
The inverse of the function y = x² - 4x is not represented accurately by the provided options, as none account for the ± when taking the square root in the process of finding the inverse. The closest option though is (a) y = (x - 2)². Option A is correct.
Step-by-step explanation:
To find the inverse of the function y = x² - 4x, you need to switch the roles of the variables x and y and then solve for y. The steps are as follows:
Replace y with x and x with y to get x = y² - 4y.
Bring x to the other side of the equation: x + 4y = y².
Rearrange the equation to the standard form of a quadratic equation: y² - 4y - x = 0.
Complete the square on the left side: (y - 2)² - 4 = x.
Finally, add 4 to both sides to isolate (y - 2)²: (y - 2)² = x + 4.
Since we need y by itself, we take the square root of both sides, getting y - 2 = ±√x + 4.
Lastly, isolate y to get y = ±√x + 4 + 2.
From the options provided, the one that comes closest to these steps is option (a) y = (x - 2)². However, this ignores the ± from taking the square root. In reality, the inverse function is not a function in the strict sense as it does not pass the vertical line test due to the ±. It should be separated into two functions if we want a one to one mapping.