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Find the inverse of the function y = x² - 4x.

a) y = (x - 2)²
b) y = (x - 2)⁻²
c) y = (x + 2)²
d) y = (x + 2)⁻²

User Afrischke
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1 Answer

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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.

User Michael Samuel
by
8.0k points
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